Can running Firefox and Chrome at the same time cause problems?
Sometimes when I switch from FF to chrome, where I have many more tabs up, the pages on chrome refresh slowly (1 second?), from top to bottom.
The only problem here could be running out of memory or network congestion from having a data transfer intensive page open in the other browser (e.g. youtube video playing or streaming audio). Running more than one unrelated web browser at a time is certainly not a problem otherwise.
Short answer (all memory and processor requirements aside): No.