Towel in vent is it safe?

I would not do it. It could eventually dry out and catch on fire.
Vent covers are usually not very expensive so I would invest in one and flip it off.

Welcome to Fluther!

Welcome to Fluther.

There is absolutely no fire danger from using a towel to plug a residential air heating vent in your ceiling. There is no risk because the air temperatures in a residential heating system won’t even begin to approach the ignition temperature for the fabric. Or paper or cardboard, for that matter.

However, plugging the vent with anything will accumulate a lot of dust and may also become a home for rodents, insects and other undesired creatures. Your best bet is to keep the vent open, but to redirect the airflow to a more desired location, or to find the damper in the duct – if there is one – to close it or partially close it to restrict the airflow that way.

Should be safe if heating system is not on. Remove it when you turn the heating unit on. Be sure it is an outlet and not an inlet (cold air return). Like @chyna said, vent covers aren’t too expensive, usually.
Are you trying to keep out the heat of summer or the heated air in winter??

Yes it’s perfectly fine.

No. Buy a grill to put over it for heaven sake. If you rent demand the landlord do it. You could have a rat enter your residence through the duct if you live in an old apartment building. A temporary safe (I believe) thing to do is buy an HVAC disposable filter and put over the vent hole.