I want to remodle my basement and maybe put a pool table down there. The issue I have is that there is a support post in the middle of the room. The room is 16'x24' with the wooden 2x10 beam being 16' in length. My question is, what size I-beam or comparable beam would I need to eliminate the post. The house is only 1100 sq ft. Ranch "Bungalow", no attic.
Any suggestions?

In a forum such as this, there is too ,much risk in offering structural design services, because one can never trust the information one is given. The best person to answer your question is a qualified design professional who can examine the house first-hand, and discover any existing conditions that would affect the structural design.

Then there is an issue of the foundation being adequate to handle all the new loads at the ends of the beam you might install. It can't be assumed that whatever is under what you see is sufficient to carry the new loads. There is also the problem, sometimes, in getting the new beam into the house and in place, that has to be solved.

It isn't as simple as it sounds, and further, people like myself make our living doing such design work. Why should I give it away for free, and possibly based on faulty or incomplete information. at that?

If you are serious about doing this call a structural engineer and have him come look at it and do a write up and post installation inspection. That 2x10 is probably a triple 10 or even a lvl.

...or even a decent architect. It isn't rocket science, but it does need someone to examine the house first-hand. I often do stuff like this free of charge for past or present clients, or at minimum cost for others.