It is entirely possible. Here is one of many videos on YouTube.
Unfortunately, it is impossible to predict how much time or warning you will have before a car fire can occur. If you don't have the money to fix it now you should at least make certain that oil isn't allowed to drip directly onto the exhaust. I know the video may seem like the extreme possibility but, that is what you should keep in mind when dealing with flammable fluids and materials.
If oil leaks on your exhaust, it can be dangerous. The buildup of oil
on your exhaust can potentially cause a fire. A car fire is something
that nobody wants.
If the leak is big enough it can burn that sucker to the ground!
(Been there , done that. 350 Chevy C-250, summer ,1981)
...check for oil leaks and always use a funnel when adding oil. Oil
spilled on a hot exhaust manifold can cause a fire.
If you have a
gas station attendant add oil, double check that the cap is on
securely. This sounds obvious, but better to check than end up with
oil all over your engine compartment at best, or an engine fire at
Another video on youtube.com
The flash point of Mobile 1 10w30 motor oil is 232 Deg. C. according to its data sheet.
Exhaust gas temperatures can range widely depending on the car. An average I've found mentioned is 149 Deg. C. at idle to 760 Deg. C. at high load and rare extremes of 1200 Deg. C. Below is a quote by "the solitaire"from sportscarforums.com
EGT (exhaust gas temperature) can easily range between 300 degrees F
at idle and 1400 degrees F at high load.
The 232 Deg. flash point of motor oil easily falls within the range of exhaust gas temperature.