What you ask for might be possible if your planet and another planet or star in the system have a long enough synodic period.
And it might also be possible if your planet and another planet are co-orbital. How could another planet help to heat up your planet? That will be explained in the long answer.
Part One Of Nine: A Complaint.
Your title is misleading. The planet doesn't orbit around "two binary stars" which would be a total of 4 stars in two pairs, it orbits around a single "binary star" which is one pair of two stars, or it "orbits around two stars which are a binary system".
Part Two Of Nine: Problems with a Habitable Planet with a Long Cycle of seasons.
There are big problems with having a habitable planet in an orbit which takes about twenty Earth years. The question didn't say "Earth years", just "years" But since one orbit by the planet is one of that planet's years, one orbit can not possibly equal 20 years of the planet. So I assume that you mean Earth years instead of Mercurian years or Neptunian years.
For a planet to be habitable - the question doesn't specify that, but since it is a science fiction story I assume that the planet should be habitable for liquid water using lifeforms in general or even for humans in particular - it has to have a relatively steady inflow of radiation from it's star or stars, and for a very long time, billions of years.
As far as I know, the main scientific discussion of the requirements for a planet to be habitable for humans is in Habitable Planets for Man, Stephen H. Dole., 1964.
Chapter Four: the Astronomical Parameters, starts with a section describing the properties of a habitable planet. The proper age of a planet is discussed on pages 61 to 63. The plant Earth didn't becme habitable for humans until about four billion years after it formed.
Dole concludes with:
In general, it is probably safe to say that a planet must have existed for 2 or 3 billion years, under fairly stable conditions of solar radiation, for it to have matured enough to be habitable.
And there is a section about Properties of the Primary (the star) on pages 67 to 72.
Stars shine with fairly stable amounts of solar radiation durin the main sequence phase of their "life cycle". After the main sequence phase the luminosity of a stars has drastic changes which should kill all life on its planets - in some cases the star will actually totally destroy some or all of its planets.
As a rule, more massive main sequence stars usually have much more hydrogen to fuse into helium during their main sequence phases. But their luminosities, the rates at which they emit radiation (and thus the rates that they use up their hydrogen to produce that radiation) increase at a higher rate than their mass. Thus more massive stars spend shorter periods of time in the main sequence phase of their existence.
On page 68 Dole says:
The only stars that conform with the requirement of stability for at least 3 billion years are main sequence stars having a mass less than about 1.4 solar masses-spectral types F2 and smaller-although the relationship between mass and time of residence on the main sequence is probably not known with great accuracy and is subject f to futue revisions (see figure 25). Stars havng masses greater than 1.4 solar masses spend less than 3 billion years in the main sequence and then go into evolutionary phases in which the energy ouput changes rapidly.
The answer by user177107 to this question
Has a table giving the characteristics of stars of various spectral classes. It also give the calculated orbit of a plenet receiving the same amound of energy as Earth receives from the sun, including the length of the planetary orbit in Earth days.
For a F2V class star the orbital distance would be 2.226 Astronomical units or AU (the distance between Earth and the Sun) and the planetary year would be 1,018.01 Earth days, or about 2.78 Earth years.
And perhaps that might be the longest possible year for a planet that receives exactly as much radiation from its single star as Earth gets from the Sun.
But the star is supposed to be a binary star so the planet gets radiation from two stars that it orbits around.
If we make both of the stars F2V class stars, the distance to an Earth equivalent orbit will be increased by the square root of 2, which is about 1.141. So the Earth equivalent orbit around two F@V stars would be 1.141 times 2.226 AU, or 3.148 AU. And with an orbital radius of 1.141 times, it will also have an orbital circumference 1.414 times as great. So if the planet orbited at the same speed, its years would be 1.414 times 1,018.01 Earth days, or 1,439.466 Earth days, or3.941 Earth years.
But doubling the mass that the planet orbits must also increase the required orbital speed. Thus the length of the planet's year will not increase by as much as 1.414 times. Without doing the calculations, I can't even predict if it will increase at all.
I also note that since the two stars are orbiting much closer to each other than the planet is to the pair, the orbital periods of the stars around each other will be only a fraction as long as the orbital period of the planet around the two stars.
Thus even if you could get the planet to have an orbit around the stars that was twenty Earth years long, the alternation between the brighter star being closer to the planet and the dimmer star being closer to the planet would happen several times during the twenty Earth year orbit of the planet. Thus there would be several complete cycles though the seasons during one twenty year planetary orbit.
Part Three of Nine: Suggestion One:
It would be easy enough to give a planet a year 20 Earth years long. Put it in orbit around a spectral class O or B star where it would have an orbit 20 Earth years long in the star's habitable zone. The planet would have to be very young and terraforemd to be habitable by an advanced civilziation. Or else it could be an old planet that naturally became habitable and was moved from it original star to orbit around a hot young star.
But if the planet orbits around two stars, for reasons of stable orbits the distance between them has to be small relative to the distance at which the planet orbits the pair, so that they will make several full orbits during each planety orbit, and thus alternating periods of more and less stellar radiation, and thsus planetary seasons, will happen several times during the orbit of the planet.
And that is true regardless of which spectral type of stars the planet orbits and what its orbital distance is.
The same will also happen if a planet orbits one star and there is another farther away in the system. The two stars will be far enough apart for reasons of orbital stability that the planet will make several orbits around the near star for each orbit of the two stars around each other.
One way to handle that problem would be to make the orbits of the two stars rather eccentric so at times they get much closer, and make the farther star much more luminous than the nearer one, thus creating summers when the two stars are closest together. I think that I discussed the mathematics of such a situation in an answer to another question.
Possilby ths one:
Could an Irregular Orbit Cause Significantly Longer Seasons?
Part Four of Nine: Synodic Periods.
Another method would be to use the synodic period of the planet and one of the stars instead of the orbital period of the planet.
The synodic period of two objects that orbit a third object is the time it takes for the two orbiting bodies to return to the same relative positions again. For example, it could be the time between two consecutive moments when two planets are on opposite sides of their star, or the time between two consecutive moments when the two planets are lined up on the same side of the star and approach each other the closest.
So the synodic period of two objects orbiting the same object is a mathematical relationship between how long their orbital periods are relative to each other.
And here is a link to an article which has a curve showing the lengths of the synodic periods of solar system planets relative to Earth.
Note that the greater the difference there is between the orbital period of a planet and that of Earth, the shorter the synodic period will be, while the smaller the difference there is between the orbital periods of the planets the longer the synodic period will be.
So if the second star in the system orbits just within or just outside of the orbit of the planet, the synodic period will be many times as long as one orbit of the planet around the first star. By making the difference between the orital periods of the second star and the planet small enough, the syndoic period between the second star and theplanet can be made arbitrarily long, which includes a synodic period of 20 Earth years if that is desired.
So if the planet orbits just inside or just outside the orbit of the secondary star, the planet would get farther and farther away from the heat and light from the second star during half of their synodic period, and in the other half of their synodic period it would get closer and closer to the head and light of the second star. So for part of the synodic period of the second star and the planet the planet would be close enough to the heat and light from the second star and hte planet would heat up significantly.
Part Five of Nine: The Flaw in the Synodic Period solution.
Except that as I stated earlier in this answer there has to be a big difference in the orbital distances, and thus the orbital periods, of the planet and the second star for their orbits to be stable over billions of years. If the planet and the second star obit the first star in orbits which have similar radii and thus similar orbital periods, and thus have a long synodic period, the system will be unstable and the planet, being much less massive than the second star, will be the one that is ejected from its orbit - which will be very bad for life on the planet.
Part Six of Nine: The Synodic Period with Another Planet instead of Another Star.
The only way to avoid such a problem would be to put the planet in a orbit very close to the orbit of another planet orbiting the planet, thus giving the two planets a very long synodic period, which can easily twenty Earth years long if you wish.
But how will the planet be heated up by the other planet when they pass close once every twenty year long synodic period? Planets are notorious for only reflecting light and heat from their stars and not generating any heat and light of their own.
So the planet which acts like a smaller sun to your planet will have to extremely hot for a planet, glowing red hot, to emit enough heat and light to warm up your planet during close passes.
Part Seven of Nine: How to Make a Planet hot Enoough to Act Like a Star.
So maybe your planet formed in a different solar system and gradually became a habitable planet over billions of years. Its star began to swell up into a red giant which would soon kill all life on the planet. But a super advanced civilization moved the planet from its original star into orbit around another star - a very young star with very young planets. The planets in the system had formed so recently that they were still glowing red hot from the heat of their formation. And the advanced civilization moved the planet in your story to an orbit very close to the orbit of one of the glowing red hot planets, which gave the two planets a synodic period of about 20 Earth years. And once every synodic period your planet would pass close to the red hot planet and be heated up by its visible and infrared radiation.
And that would continue for millions of years as the young new planets gradually cooled off.
Or possibly the planet in your story is in your its original solar sytem and all the plannets are billions of years old and thus cooled down to normal temepratures billions of years ago. Your planet's orbit is very similar and close to that of another planet, so they have a long synodic period and pass at their closest once every synodic period. And the two planets look very spectacular in their skies during the close approach, but they don't heat each up up much.
In the early periods of star systems, planets form in unstable orbits, which gradually become stabilized as the planets with the most unstable orbits fall into the star, are ejected into interstellar space and become rogue panets, or else collide with other planets. And after a few hundred million years, almost all of that is over, and the remaining planets are in orbits which will be stable for billions of years. But it would still be possible, though rare, for planetary orbits to be destabilized and two planets collide, even after bilions of years when that never happened.
So perhaps after billions of years without any orbital problems, minor perturbations finally add up to enough to force a planet in the system out of its orbit into a new orbit which puts it on an eventual collison course with the neighbor planet to your planet. the two planets collide and are shattered into incandescent vapor and red hot lava, and most of the pieces gradually reform into a new planet with an orbit similar to that of the neighbor planet. Thus the new planet also has a synodic period of about 20 Earth years. And now, whenever there is a close approach to the new planet, it emits a lot of light and heat and warms up your planet, a situation that should continue for millions of eyars during the cooling off period.
For close approach to a red hot planet to work, you will want the two planetary orbits to be very close in relative terms, to have a very long synodic period, and also to be very close in absolute distance so the close passage is close enough for the red hot planet to warm up your planet signficantly.
Part Eight of Nine: Can Planetary Orbits be That Close?
The planetary orbits in the famous TRAPPIST-1 system are quite close in both relative and absolute distance.
The orbits of the TRAPPIST-1 planetary system are very flat and compact. All seven of TRAPPIST-1's planets orbit much closer than Mercury orbits the Sun. Except for b, they orbit farther than the Galilean satellites do around Jupiter, but closer than most of the other moons of Jupiter. The distance between the orbits of b and c is only 1.6 times the distance between the Earth and the Moon. The planets should appear prominently in each other's skies, in some cases appearing several times larger than the Moon appears from Earth. A year on the closest planet passes in only 1.5 Earth days, while the seventh planet's year passes in only 18.8 days.
The orbital period of planet f is 9.206 Earth days, and the orbital period of planet g is about 12.353 Earth days, giving a ratio of about 1.341, the smallest ratio of orbital periods in the system. According to this synodic period calculator:
The synodic period of f and g is 36.137 days, which is 2.925 times the orbital period of g and 3.948 times the orbital peirod of f.
So for two planets to have a synodic period of 20 Earth years and have relative year lengths in that same proportion they would have years that were 5.06 and 6.837 Earth years long.
And as I calculated much closer to the beginnng of this answer, habitable planets probably can not have orbital periods that long.
And if their orbital periods are that long and their orbits are that wide, the absolute distances between the two orbits is likely to be so great that even a red hot planet would not heat up its neighbor much.
If the synodic period was 99 times the period of the outer planet, and 101 times the period of the inner planet, and equalled 20 Earth years or 7,305 Earth days, the orbital period of the inner planet would be 72.326 Earth days long and the orbital period of the outer planet would be 73.787 Earth days long.
Using the synodic period calculator using orbital periods of 72.326 and 73.787 days, the calculated synodic period is 3,652.785 days or 10.00078 Earth years.
If the synodic period was 999 times the period of the outer planet, and 1,001 times the period of the inner planet, and equalled 20 Earth years or 7,305 Earth days, the orbital period of the inner planet would be 7.297 Earth days long and the orbital period of the outer planet would be 7.312 Earth days long.
Using the synodic period calculator using orbital periods of 7.297 and 7.312 days, the calculated synodic period is 3,557.044 days, which is actually shorter than the previous example..
Using periods of 98.5 and 99.8 days, I get a synodic period of 7,028.678 days, or 19.24 Earth years, which seems close enough to 20 years.
Using periods of 49 and 49.33 days, I get a synodic period of 7,324.758 days, or 20.054 Earth years, which seems close enough to 20 years.
Using those shorter periods, the planets can be closer to their star and thus closer to each other, making it easier for a red hot planet to heat up your planet when they pass close to each other.
And no doubt there are many other possible combinations of orbital periods resulting in a synodic period of 20 years.
But those arrangments require the two planets to have orbits which are relatively much closer than any known examples. Is it possible for two planets to have orbits so close together both relatively and absolutely?
Part Nine of Nine: Co-orbitals.
Actually it is possible for two astronomical bodies to be co-orbital, sharing the same orbit though at different positions in it.
Theoretically it is possible for two objects to share almost exactly the same orbit, with only a slight difference in the semi-major axes of their orbits. Thus they have very long synodic periods. And when the slightly inner and slightly faster object finally catches up with the slightly outer and slightly slower object, their gravitational interactions causs them to change places, so the former inner is now the outer and the former outer is now the inner.
Astronomers know it is possible to have a reasonably stable orbit like that, because one has been discovered decades ago.
Epimetheus and Janus are two moons of Saturn which share the same orbit in such a configuration. The differenc in the semi-major axes of their orbits is only about 50 kilometers, which is smaller than their radii. Their orbital periods are both about 0.694 Earth days, with a difference of about 26 seconds in orbital period. The synodic period of the two moons is about 4 Earth years, or about 1,461 Earth days, or about 2,105 orbits.
If two planets had a synodic period of 20 Earth years or 7,305 Earth days which was about 2,105 times as long as their orbital periods, their orbital periods would be about 3.47 Earth days long with a difference in length of about 130 seconds or 2.16 minutes.
If two planets can have orbits that close, it might not be necessary for one of them to be composed of red hot lava. When the two planets are orbiting almost side by the side, the other one will reflect a lot of light onto the inner one, and their tidal interactions should create a lot of tidal heating. And they should be travelling close together for a signficent percentage of their total synodic period, and the summers caused might last for a significant time.
And that is even more the case if the two planets actually switch orbits once in each synodic period.
As they get closer and closer planet A may be the inner planet and get more an dmore heat reflected from t palnet B, the outer planet, and the tidal heating on both of the planets should get stronger and stronger. At the moment of orbital exchange the tidal heating should be at its strongest, and then it should gradually diminish over weeks, months, or maybe years as the planets gradually separate. And after the switch, planet B will now be on the inner orbit and get a lot of refelected ehat from planet A which will now be the outer planet.