I'm trying to have an upgrade ability, upon completion, done by either player 1, 2, or 3, upgrade the towers of player 4.

I know how to make requirements and upgrades for someone to upgrade their own buildings, but I'm not sure how to do it so that one ability, used by one player, will upgrade someone else's (a computer in this case) buildings.

I think the easiest way would be to use a trigger, which would be something along the lines of

Event
Player uses ability <x>
Conditions
Ability = Upgrade!
Triggering Player == 1, 2, 3
Actions
Set Upgrade <x> to level 1 for player 4

I don't have a copy of the editor on hand, unfortunately, so I can't give you the actual code, but i hope that will give you a push in the right direction.

I was trying to work out something like that before (I was hoping it would be straight forward like what you put up), but I don't see most of those options. There is only "unit uses ability", but the "unit" is from a building that is going to be built in game, so it's not already on the map, I also don't see any options for conditions like that.

There are 4 different research buildings that can be built, depending on the stage of the game. Each person researchs what they want independently. Each research building has about 15 different upgrades that do different things, 2 of which in each building that (I want to, regardless of who uses it first) cause all the starting towers to upgrade. There are no upgrade levels or anything like that. The towers (which can be built by players, but there are also a few that start out on the map) have morph abilities on them.

The reason I need any player to be able to upgrade all starting towers is because (as an element of the game) there will be one person teching, so that would be an element of their job. The starting towers might need to be different buildings from what can be built by players for it to work, but that wouldn't be a big deal to do.

I found a sort of workaround (that will have to do for now). I made a tower upgrade building for each player and buttons that have requirements attached to them (requirements of the research abilities that unlock towers). I then made a trigger that has it so that whenever the button is pressed by players 3,4,5 (team 1) it will kill all of the starting towers of player 1 (computer allied with team 1) and create the next tier tower in that same place. Then I did that for the other team and each tower tier.

Well, from what I read, it sounds like you want to use a unit group of some sort.

Using actions, you can filter down what units you want to be affected when the trigger executes. "Pick Each Unit in Unit Group Matching Condition" may be the action you want to use. You can select all units of a specific unit-type, player-owner, etc. (Basically, all units you want to affect), and then run the necessary actions using "Picked Unit" to refer to each unit in the group.

So you could grab all of player 4's towers that you want to upgrade and make them upgrade to the next tier.