Counter offers can burn 2 bridges
First you are burning a bridge at your current company because you are a traitor and second the company that offered the job you are turning down to remain may no longer want to hear from you again either. I would only accept a counter offer with a multi year can't be laid off or fired contract. If they won't give that (and almost no one will) you should never accept the counter offer. They are just buying time to dump you either its finish a project or train a replacement.
Usually the reasons for wanting to leave a company have a little to do with money and mostly with conditions/working with bad people. Money does not fix the second, its always better to stick to your guns and move on.
