I currently drive a 1990 Oldsmobile Cutlass Ciera. Today while driving it I noticed a few wisps of smoke coming from the car when I started it up. There was also a smell coming from it as well, not like antifreeze, oil, or any other liquid I know of. It seeming to have a burning hair or meat smell to it. The car also did not smoke while driving it, just on start up. Could I possibly have a rodent or other small animal stuck in the vents?
If the smoke came from the dash board inside the car, I wouldn’t play around with it. If an electrical fire starts under the dash, the whole car will be in flames before anyone can stop it. Get it checked out ASAP.
Only problem is that the smell has done this before, off and on. Nothing consistent really, it will sometimes happen and sometimes not. The smoke was a new thing.
Does the smoke come from out of the vents on the dashboard?
Tester
Nope, it looked to come out of the seams. And was only a few wisps.
Have you ever seen what can happen when all the wiring shorts out under a dashboard? I have.
A friend was leaving, got into his car, and as I was waving goodbye at the windshield, and he was smiling waving back, he hit the key to start the engine, and in an instant the entire interior filled with gray smoke where I couldn’t even see him thru the windshield. The door blew open and he bailed onto the ground coughing and hacking.
So the smoke you see now may be a prelude of what’s to come.
Tester
You might want to have that smoke “leak” checked, it is hard on a car to drive with the smoke level low 
Really get this looked at ASAP, as even a minor amount of electrical system work would cost so much it would effectively total your car.
A fire in a car is nothing to screw around with. And smoke means fire. The whole car can go up in minutes. Rear the dashboard apart and start the car . . . with a fire extinguisher nearby . . . and find the cause. I saw a car burn up in less than 5 minutes . . . really scary! Rocketman
A rodent probably partially chewed some wires. Without tearing out the whole dash, you’ll never know.
Burning alive in your car is a very real possibility. A crash could be the event that provokes a full blown fire.
Nothing short of a charged 1.5" fire hose is going to put that fire out once it gets started. If it breaks out when you are going 70mph, it might engulf the whole dash before you can come to a complete stop. These things spread scary fast, and they are deadly.
As a former EMT, I had the misfortune of seeing a couple of vehicle fires with entrapment. The sight and unforgettable smell give me nightmares to this day about being burned alive. I’ve seen a lot of people badly hurt, but burns are the worst. And no family anguishes more than those who have had a loved one burned alive.
I urge you to take this very seriously and disconnect the battery right now. Tow it to a good mechanic. This is a big deal. Where there is smoke, there is soon to be fire. It would blow your mind to see it. It happens in seconds.
Is it possible the resistor that regulates the heater fan speed is overheating and smoking? As others suggested, I’d get it checked out soon! The interiors of cars are made of really flammable stuff that goes up like you wouldn’t believe if you’ve never seen it.
If you aren’t the original owner, it may be worth checking the fuse box to make sure all the fuses are the right value too—I had a used car on which someone had replaced every fuse with a 30A fuse because some kept blowing. I found a wire under the dash burnt in two and tied together in a bow! Ultimately the problem was a ground strap that was dangling from an incompetent radio installation and randomly shorting things out.
If you plan on gambling with the car though, I definitely wouldn’t park it at night in your garage or near anyone else’s vehicles or anything flammable.
Asking an online forum what to do when smoke is emanating from the dashboard is sort of the automotive equivalent of asking firstaid.com what to do when you have crushing chest pain. In both cases, the appropriate thing to do is to walk away from the computer and go to a qualified expert right away before something much more serious transpires.
As has been said, the smoke condition is very likely the prelude to a conflagration. This is not something that can be ignored while continuing to look for online advice.
Disconnect the battery and have the car towed to a mechanic before you lose it (and possibly your life) as a result of a car fire.