Can I just tape over the seam where the duct meets my furnace?

You can go ahead and tape that seam without worrying. Just make sure you use proper **UL 181 aluminum foil tape**—the stiff metal stuff with the peel-off paper backing—rather than regular cloth duct tape, wich turns brittle and falls off after one winter of heat cycles. Give the metal a quick wipe with rubbing alcohol before sticking it down so dust doesn't ruin the adhesive. The only thing to watch out for is that vent pipe. If it's a metal B-vent or single-wall exhaust pipe, keep the tape a few inches away so the radiant heat doesn't degrade the adhesive. Other than that, the seam between the furnace cabinet and the supply plenum (or cased coil) isn't an air intake or an exhaust. You aren't suffocating the burner or trapping carbon monoxide. Just be careful not to tape the furnace door shut, since a tech still needs to pull the front panels off for tune-ups and filter checks. Installers skip sealing every last inch mostly out of speed, but also because that section right above the furnace usally houses the AC evaporator coil. If a tech needs to get inside that coil box to clean fins or check a leak, peeling thru mastic or layers of tape is a pain. Besides that, a few cubic feet of air leaking into a conditioned basement or utility closet doesn't really move the needle on your electric bill, so a lot of crews just tape the rough spots and call it a day.

Seal it up if the draft bugs you. It takes two minutes, keeps the air heading toward your registers instead of the utility room, and won't hurt the system at all.

Quick question about the coil access panel—I'm totally new to HVAC stuff and can barely tell the parts apart. How do you tell the difference between a removable door and just a normal fixed seam? My setup looks like one giant tin box sitting on the furnace, and I'm terrified I'll tape over something a tech has to pull off later.

Also, dumb question, but if you seal up leaks like that, does it actually push noticeably more air up to the colder rooms, or is a gap like that way too small to matter?

Look for 5/16" hex head screws—those outline the removable coil door. Fixed plenum transitions usually just have S-cleats or drive slips with no intention of coming off easily.

Honestly, even if you tape across a service seam, any tech worth a dime will just slice through it with an Olfa knife in two seconds flat. Pick up a roll of Nashua 324A or Shurtape AF 100 and press it down flat with a squeegee. And don't sweat the airflow numbers—patching a tiny 10-20 CFM leak won't bump your total external static pressure past your target 0.5" w.c. Just seal it.

Totally agree on using proper foil tape there, but make sure you shut off the power switch on the furnace before you stick it down.

If the blower kicks on while you're trying to lay the tape, that positive air pressure will puff the strip right off the metal or cause bubbles under the seam before the adhesive even grabs. Much easier to get a flat, airtight seal when air isn't actively fighting you.

This is blowing my mind, I literally just ran down to my basement with a flashlight to check mine and found like three spots puffing warm air! Quick question on that though—if fixing leaks pushes more air upstairs, is it safe to use those magnetic vent covers on registers in rooms that get way too hot? Or does blocking vents mess up the furnace pressure? My home office is an absolute sauna while our bedroom freezes!

Quick update — cutting the power switch first helped a ton with getting the UL 181 tape stuck down flat without it bubbling up from the airflow. But the gap near the back corner turned out to be way wider than I thougth, almost a quarter inch, and the tape keeps sagging into it. shold I double up multiple strips there, or is mastic paste the better fix for a wider gap?