I have a MVP system setup where the units are transferred to player Orange to player Brown. Each time this happens though, the units un-burrow. Although, for player Red to player Brown the units do not un-burrow. The only difference is that I (Human) is player Red and the other two are AI's.

I do have burrow researched for all players by default.

Maybe it has something to do with computer players that own no units get their AI cancelled.

I'll do some testing with more humans in the game and see what happens.

Yes, this has something to do with computers' AI, as Nude have said.
There're two solutions.

1. Make brown Neutral, rather than a computer.
2. Disable brown's burrowing ability. As long as no burrowed units are supposed to be created by that player, you can give burrowed units to the computer.

I have not tinkered with AI, but there MIGHT be an AI setting that is similar to being Neutral for the computer. But if that's the case, you might as well set the computer to neutral.
You can also try P9-11. These will automatically attack anything they're enemies with but do not have AI.