furnace blower motor vibration when working hard

Try putting pressure on random things to isolate the location. Take the top panel off and start there. I'm not sure how you can check the rest if it doesn't do it while you have access open to deaden random things.

Try tightening the motor mounts and the blower cage bolts. Then do every screw you can find. A last resort might be shoving toothpicks in between where two pieces of sheet metal meet. Newer blower cages only have bolts on the front, the rear rests in a slide; another possibility.

If it's the bearings in the cage or the motor there's not much you can do except replace them, assuming you can verify either of these as the culprit.

Just as with car wheels, vibrations happen at certain speeds. Putting the filter on, no matter how clean it is, slows the motor down a bit, which just happens to be an rpm where any out-of-balance vibrations show up (in your case).

It has been my experience that when there are excess blower vibrations, the cage blower itself has accumulated dirt. They can be in perfect balance, even though coated with dust, till a chunk of dust falls off from one spot, thus throwing the whole thing out of balance.

Clean the blower with compressed air (after removing it) and see if that helps. Note: do not disturb any balancing weights that might be attached to the blowercage.

Your initial assumption about load on blower due to new filter is valid. Called starving the suction of the blower. If the MERV rating of the filter is too high, this will starve the blower suction and cause any minor vibration in the blower motor/wheel assembly to amplify. In my case the imbalance was due to a failing motor bearing.

I read this post last night when one of the blowers in my system started vibrating, shaking the entire house. The comments above encouraged me to look into it myself, and I wanted to share what I've found in my case for the benefit of anyone in a similar situation.

In my case, it seems the Installation Instructions were left inside the unit and pages of it got jammed on the blades, causing the imbalance. After I removed these, everything went back to normal. The AC is 6 years old, so these instructions must have been there since the installation, probably taped to a side panel internally.

Some things you can do to calm annoying vibrations:

You could try just cleaning the flywheel. Of dirt. Might help.

Install a duct "skirt" "canvas duct connector" so that vibrations don't shake the structure as much.

You can try to "rebalance" your flywheel. Or replace it. Or replace it and the motor. Decreases vibrations.

You can add some padding underneath the furnace. A lining around the underside perimeter (if it's downdraft), or similar.

Could also add padding between the ducts and the wall.

If you want to be really aggressive, replace the furnace with a "two stage blower" then as long as it's close to temperature, it won't blow as hard. Or even variable speed even more so. But more expensive.
Some "normal" furnaces have an option for how fast they blow either "high high" or "high medium" so youc an vary the speed of the blower down a bit.

Or install radiant floor heating instead of forced air.

If your furnace hangs from the ceiling there are little rubber pads they can use to hang it with "compression fittings" that reduce vibrations.