New to this, I have a question about assets

When building assets what are the recommended maximum number of Tris in an asset?

In particular, for an MMO Avatar? Weapons? Buildings? Props? Other players? Pets/Mounts? Monsters/NPC?

This depends on your project CPU / GPU budget, which in turn depends on your target minimum hardware, as well as what is going to constrain your game, and how much CPU you intend to devote to game logic, and GPU you intend to devote to advanced shaders.

If you want your game to run on a GMA 950, your poly counts will be very different than if it only needs to run on an XBOX 360.

If you only have a rotating cube on screen, but highly complex AI logic, it really doesn't matter how many vertices your cube has (to a reasonable limit).

The problem with guessing, is that you may end up wasting time optimizing art, when it isn't really benefiting the overall project.

It sounds like you are intending to work on an MMO, in which case, I would recommend defining a performance budget (x milliseconds for AI, y MS for rendering, etc) and testing prior to doing a lot of work. (This is the cue for someone to jump in with the standard MMO warnings).

If your project allows it, I'd also recommend including dummy art (cubes, spheres, other primitives) of approximate poly counts until you have a better idea of performance constraints, and get further into the project.

Of course, failing that, you can always copy existing games poly counts, with WoW being a pretty safe MMO in terms of not requiring very advanced hardware. Just keep in mind most AAA games write their own game engines from scratch, which allows them to optimize in ways you cannot if you use an existing engine.

I believe their characters are between 1-2k. The buildings and trees are generally so low poly, you could probably count the polys in game, and large bosses come in around 5-10k.

When you test, keep in mind both the total poly count, and number of objects on screen will affect render time. So if you intend to have 100 people on screen, test with 100 objects at your desired poly count, on your desired hardware. With some basic math, you can get an idea what your rough range should be.

If you really have no idea yet, you can always make a few versions at various poly counts (LOD) and swap them out at will.

In addition to above, don't forget about texture budgets either. These will take up the majority of your video ram budget, and can add to rendering complexity. If using Post Processing shaders with Render Targets, this will be especially important as any effects using the full screen resolution as target texture size will eat more of this budget as the end user increases resolution.