How should I seal a butt joint in a tight space?
I am sorry to hear about your water damage issue - stinks for sure.

You mention if it were adhesive you could do it from either side of the hvac - sounds like you have access to the back side of that unit ?? If So see method 1 if not see method 2.

METHOD 1:

Use the dry wall tape on a flat stick and push to the back side so you can get to it. Then use Push Pins (like thumb tacks) to hold it in place. Pull it up into position on the front side and secure it with pins - you can then run some joint compound over it a long handled putty knife (probably need to make your own handle here) or use a 'paint roller' to try and spread it flat.

METHOD 2:

To repair that tight space dry wall butt (amazing you installed the dry wall there). What I would do is make a jig that can hold my dry wall tape on one end like a long flat bar (much like you have sandpaper in a small sander , place joint compound on the end and slide that in place and get the end in the correct position. Set your jig in the proper position that the tape would be able to be pulled across flat . Wait for the joint compound to dry - now it is secured to the back side and you can add compound with a roller or a long handled putty knife while pulling the tape tight and flush.  

METHOD 3:

Some drywall tape actually does have an Adhesive Backing - look for the WEB TAPE with Adhesive..



EDIT 8 MAR 2018