Dodge neon overheating

I have a 2002 dodge neon that overheated
due to the heater core going out. My uncle
bypassed the heating system to prevent
antifreeze from leaking and so I could drive
the car without it overheating. I thought all
was well as its driving fine, a little shakey but
its always drove that way, until I've noticed a
burnt rubber type smell. I didn't know if I
was being paranoid or if something really
was wrong so I had my aunt come out and
she said it smelled like it was about to over
heat. My temp gauge is broke so I don't
really have anything to go off to tell if it's
going to overheat. Theres plenty of
antifreeze / water in it, no leaks. I'm having a
mechanic check it out but it's gonna be a
couple days. Wondering if anyone out there
had any ideas on what the problem might
be? I know this isn't a lot to go off but just
thought I'd give it a shot.