Jones wrote:
snip
The past couple of years, I've noticed the fan makes a "rum rum rum" sound that seems to be tied to the number of RPMs the fan is doing.
I've looked up inside the motor housing and I notice there seems to be
some dust mixed with either cooking fat or motor oil (I don't know
which) up in there, and I wonder whether the motor should be cleaned
out and/or lubricated?

No. Fans are commonly made with long life sealed bearings. When the
unit ages little can be done to prolong it's life. If you want to do
some feel-good cleaning, remove and reinstall the blades one by one and
get rid of the gunk. Go ahead and vacuum out the motor crud if that
would please you. Turn on the unit and if there is no improvement, then
you won't have a filthy fan to deal with as you install the new fan.
First check the fan mounting box and supports and upgrade if necessary.
Good luck.

Joe