Will burn have a higher chance to be inflicted. Like a water type has a higher chance of geting paralyzed than a normal type etc...?
And I couldn't find it (Okay I was lazy.)
All gens.
No it does not have a higher chance f burning a grass type. Neither source mentions it having a higher chance, which is a good indication that it does not have a higher chance.
Bulbapedia Serebii