Oh. Oh god. There is no situation in which this doesn't end badly. Not a one. Lets go down the list from gross effects to fundamental ones:
1: The sun collapses. So your sun is somehow stealing heat from things. Cool. That means that rather than radiation pressure forcing the outer layers of the sun out it's actually going to pull them in. Your sun is going to be a super cold brown dwarf. Doesn't impact your world too much, until you stop and realise that in your world space is also incredibly hot, and thus will exert further pressure on the sun to push it in. So for starters you're going to have to redefine radiation pressure. Which is a problem, because light is radiation, and your sun (at the same time as pumping out negative Infra-red) will also have to pump out positive visible light (it doesn't make things darker). This is a contradiction you'll just have to handwave.
2: The magnetosphere doesn't work anymore. In our world, the pressure on the center of the earth (along with a couple of other bits) lead to a molten core. This helps with our magnetosphere. If you keep this behaviour (given that hot->cold) then our planet will freeze solid from the inside out, exposing us to hideous levels of solar radiation from all sides (given that space is hot!). If you flip this behaviour around you get a contradiction with the behaviour of the sun.
3: We suffocate. Higher altitude gases get more energy from space. With extra energy they can achieve escape velocity and head out into the vast universal plasma, from where they will never return. The cooling effect of the sun will prevent this happening, but only in the daytime. At night-time all the atmosphere will be sucked off the planet. This can be categorised as Bad. If you reverse this behaviour then all the hot air will collect at the surface of the planet, leading to some weird weather patterns, and rapidly evaporating seas. Which brings us to our next point:
4: The natural state of all materials is plasma In this universe thermodynamics seems to be running backwards, so rather than everything radiating heat into space everything is approaching the cold-death of the universe (contradictorily this is when everything is as hot as it can be). If you keep materials behaving as we have them, then all the water will heat up and become water vapour. This steam will then sink (due to the above paragraph) leading to diving being the most hazardous profession in the world. Also a lack of fish, or anywhere nice for life to evolve. Or rain, in fact! If you take this process far enough, eventually the Earth will melt. Luckily you've got the sun cooling everything down, so the bottom of the oceans will only be a balmy 150 degrees.
5: Life as we know it can't exist. It just can't. The fundamental properties of plantlife, cellular structure and even DNA will have to be altered for a reversal in the hot->cold world. We'd have to wear clothing to keep ourselves cool and stop our blood boiling. Sadly we (as warm blooded mammals) generate heat from the inside, so the only way to cool down would be standing near a fire or sitting in the sunlight. Oh, and our feet would be considerably warmer than our heads. I dread to think what the changes to the density equations would do to intercellular interaction. Basically you've got to rewrite everything about life.
6: Chemistry doesn't work anymore. Exothermic reactions now take heat, and endothermic ones create it. Along with the changes you've had to make to the way density works I think this qualifies as stopping most chemical processes we know from being the same. 2O+H ->?. Why would these things bond? Going down to a lower energy state now requires energy to be put in (which makes no sense), leading to everything eventually falling apart into a cloud of it's constituent atoms, and eventually pushing itself all the way back up the chain of atomic binding energies to be an isotope of hydrogen or an exotic heavy metal (The Iron valley can't exist in this world).
7: Physics. Oh, poor physics... As I said in the first point, to have light working one way and heat working another, you're going to have to do some redefinition of the way radiation works. At this point you have to scrap all the laws of physics we know and write a new set. Pretty much everything I've written since then is just about various facets of how contradictions occur between what we know and how your new world would have to function. If you redefine Time such that it's running the other way, then the sun does pretty much what you've described, except it's not outputting Light, it's outputting Dark. Essentially you've got to rewrite everything up to and including the definitions of time and space.
I think your sweet spot is going to have to be somewhere between paragraphs 2 and 3. If the earth is in the process of freezing after it coalesced out of the cosmic plasma (handwave that bit), and the bleed off of the atmosphere is being slowed enough by the sun, then you can have a similar world to ours. Cold fire, sadly, would break everything again, but cold nuclear reactions might be easy enough to handwave in. Imagine how chilling Hiroshima would have been if everyone was flash-frozen...