I have home with living space on the 1st and 2nd floor and walkout basement. I am planning on renovated the basement into a family room, but I have 4 lolly columns, 2 of which I would like to remove to open up the space of 20' span. My question is what size laminated beam would I have to install to carry the load of the 1st and 2nd floor or living space with a 20' span?

You'll need to have an architect or engineer run the calculations, as it depends on factors such as how the loads from the upper floors and roof are carried down to the lolly columns that you want to remove, and to make sure that the lolly columns that are remaining can carry the additional loads. Also, you may find that a steel I-beam is a better choice.