What information do I need to size a sawn or hewn beam

You've essentially got a column buckling problem here. Factors affecting the maximum load include the geometry of the beam cross section, its material properties, and the manner which its ends are constrained. You can read up on it here.

What makes it more complicated is a possible eccentric loading condition, as The Evil Greebo points out. Now you have to worry about compressive stress and normal stress. This eBook does a good job describing the loading conditions and resulting stress under the "Maximum Stress" section.

Since the current exact loading conditions are unknown, and in addition you're changing the location the load is applied in, I think it would seek professional engineering advice. (I would, and I'm a mechanical engineer - I just don't specialize in architecture.) Either that, or an experienced contractor may be able to give you a better answer. (Shirlock, where are you on this one, buddy?)

I can provide you with the answer that comes is the result of college statics and dynamics classes.

What I can not do is provide you with an answer that carries the weight that would be provided by the stamp of a public engineer licensed to practice in the state of Florida, or your state of residence.

In all sincerity, I would ask that you not accept the word anyone who has not already approved residential construction issues in the City or County of your residence. And even then, do not take their word as accurate until you have the data in writing.

Hurricanes, slow or no insurance payments, and political inaction seem to happen at the same time.

My in-laws lived between I-95 and the Ocean, in Palm Beach County. I think the biggest difference between their house and yours, is the pool edging not being flush with the deck, and the house next door was a different color.

Realize I am guessing about everything from ASTM rating of the steel that would be used, to the hurricane wind design rating required in your zip code.

I would not blink if you were told that it needs a heavy 4 or a standard 5 inch beam. The 5 inch beam is less than 1/4” larger on each side. So we will go with that one.

I know nothing about the most recent building code and large pieces of steel around residential pools. But I will bet that 32’ bits of it weighing about 325 pounds and raised in the air, on steel, are not allowed within 6 feet of the water.

I am sure that there is now something about needing enough space to walk all the way around the perimeter of a pool.

There is likely also something about footings, saturated soil, flying debris, and the term ‘grounding’ will be used at least twice. If we add 6 feet to each side, we are well inside the bounds of reasonable.

We have now hit a 44’ clear span.

It gets a little tough to tell distance, but I am (again) guessing that this will put 1 post over your pool pump water works, and the other in front of the bathroom window. You are not going to be able to move the post closer to the pool, but I am open your thoughts.

The question is now, what are you going to do about the beam that will now overlap with your eaves. The easiest answer it likely to wrap a hollow fiberglass post around the steel one. Then paint it blue and hope it is not noticed.

Since you are changing the patio roof so much, from the materials used, the distance between the supports, the additional length, and joists now being attached to steel instead of wood. I would expect the patio roof to be seen as new construction and not repair. This will require the patio roof to be self supported. This is much better for you than the alternative of having the patio roof seen as a part of your house.

It appears from your pictures that the house side of the patio roof requires the outside wall of your house, and is even attached to it with hurricane straps. The 2x on the wall, but connected to nothing else seems to be there as an attachment point should you want to put in a flat ceiling.

To recap up to this point, removing the 2 center posts and replace it with a single steel beam spanning the entire distance, you will need to build a 4 post, self standing, cabana type roof.

It can be against your house, and the roof can be integrated into your house, but it must be able to stand on its own.

In reality the posts closest to your house will need to be far enough away from your house that they will not disturbing you house’s slab. (But you might be able to offset the footing enough that the post can be right against the house. Way too much math for right now.)

Ok, at this point, it makes no sense to build anything but a rectangle of steel, and put it up on steel legs. You will need ’notches’ to allow for both ends of the new roof, to make room for where your house becomes wider.

You will need to decide what you want the roof to look like, the notched portions will be the hardest. Thanks to this now being all steel, you can add gussets, strengthening web where needed, this will help to keep the beam size down.

We still need to deal with the new joists, roof and shingles, there are some other decisions to be made first

The costs at this point are likely in excess of the quality of life you gain in having the 2 wooden posts removed.

It would be informative to call planing and zoning, and ask if they can think of any issues with you installing about 32’ of steel, where the posts are within 6 feet of a pool. The answer would allow a few facts to get involved here.

And this is how we truly figure out what size beam you need.