There was a paper on this when portal came out, though I cannot seem to find it right now.
The end answer was Yes, portals are physically possible, but the picture is probably not. The worst hand waving has to be done in the process of making the portal. Defining why two sections of space-time would get cemented together in this way is a job for worldbuilders like us, not for scientists.
Instantaneous communication is no problem, because we could curve space-time to make the link. Effectively this would decrease the "straight line" distance between an emitter and a receiver to keep the speed of light rules in place. There would be some annoying General Relativity effects during the process of placing the portals to deal with, but I don't think it would violate anything. It'd just be nuisance.
Conservation of energy requires a bit more excitement. After all, Portal's ability to break the laws of physics in this way were legendary. Most energy is conserved without effort. Momentum in equals momentum out, and that accounts for a great deal of the potential conservation issues. However, there's always gravity.
These portals would need to interact with gravitational fields to avoid violating the conservation of energy. Whatever the underlying mechanism is for gravity, it would have to bend the direction of gravity around the portal. It would, in fact, have to bend the direction of gravity everywhere to ensure gravitational potential energy can still be modeled as a divergence free gradient field. Hopefully the mechanic would be localized, so it would have less and less of an effect as you walk away from the portals, otherwise people on the other side of the world would notice when you plopped them down. That being said, the more "evil" you are in your portal placement, the more the laws of physics will have to be "creative" to keep the laws intact.
The picture of how portals work would probably not apply. Because you simply jumped off a ledge and were carried by gravitational energy to the portal, through it, and off to the far ledge, the far ledge would need to be at a lower gravitational potential even though instinct directs us to assume that, because its "higher," its at a higher potential energy. We are used to working in fields where gravity always points down. I would expect most "reasonable" portal field implementations would not have this behavior (for a reason I'll mention at the end), so it is unlikely but not impossible that the picture would work.
Now for the hard parts
The edges of the portals will be tricky. As Peter Masiar put it, we like space time to be continuous. These sharp edges will end up being mighty torrents of quantum rage... not infinitely thin, but you'd have to spend a lot of work stabilizing them because... well... you cut a hole in the universe pulled itself through itself, and then glued it all together. What do you think would happen? Fortunately, the portal graphics clearly show this chaotic dangerous region around the edges. For your safety and the safety of others, please do not lick the edges of the portal.
The final issue would be the nastiest: the energy of creating a portal. When you drag spacetime around, the conservation of energy is STILL in effect. If you change the direction of the gravity field, you will need to effectively recalculate the potential energy of every object in the universe. If you "raise" the universe, you have to put energy into it as you make the portal to offset this potential energy. If you "lower" the universe, it will emit energy which will have to be reclaimed, or lost as gamma radiation. Don't have any tumors? Well, unless you're wearing lead lined underwear, we've taken care of that too!
So in the picture example, if you do the following:
- Rest state A
- Place the portals
- Disassemble the portals
- Rest state B
Then you MUST have put in mgh worth of energy into the portal to lift you from Rest state A to Rest state B. It looks like that's about 2m higher than before. The portal gun has to do the work... you don't get the energy for free.
Also worth noting is that the gun doesn't get to "know" how you plan to use the portals, so it probably wont get to be so efficient. It will probably have to warp a lot of gravity and "move a lot of objects." Likely it will require a nuclear power station's worth of energy to generate a consistent gravity field, and then unleash a nuclear bomb's worth of energy when you collapse it back to normal (with the difference between work in and "loud bang" out being exactly 1 Chel mass * 9.8m/s^2 * 2m. I tried to weigh her, but the scale couldn't display enough digits).
So sure, you can bend the laws of physics to make the portal work. You just can't break them.