Gas heater fan noise

The problem is clearly with the gas feed. If other portions (like other burners) work okay, the plumbing in common to those and further upstream is all fine. The problem would be the last portions to the burners.

I would start by removing the grate, lifting off the burner caps, and lift up the burner assemblies. Any foreign matter in there? If there is goo, run them through the dishwasher, toothpaste and an old toothbrush, or a steel wool cleaning pad and water to get it off.

Have a look at the igniter tips: are they clean and shiny? Do the heavily used tips look similar to the lightly used tips? 200–400 grit sandpaper is the easy fix for these. Place the burners back into position carefully, making sure each seats securely and aligns with the igniter. Place the caps back on paying attention to the alignment of the nub-like posts underneath (which keeps them from sliding off) so they don't cause a gap by making the cap sit up on a post.

If that doesn't renew performance, it is either a valve or plumbing problem. I would call an appliance repairman at that point for his/her knowledge of the exact problems with that model as well as the range of solutions and prices.

The blower very likely does spin at different speeds when heating vs. cooling. This is not uncommon at all. In heating mode they typically are on a low speed, while in cooling or fan mode they are typically on high speed.

As such, it could certainly be the case that there is some imbalance in your blower that excites a harmonic frequency of your furnace at low speed, but when the blower is turning faster at high speed, no sympathetic vibrations are excited; hence vibrations with heat, but not with A/C.

As others have said, it may also be the case that you have an induced draft blower for your gas furnace that is causing the vibration, but these are typically much smaller blowers than the "main" blower.