One anecdata: The one time I accepted a counter-offer (but not for more money), I regretted it.
(I was at place that had an existential problem, and unhappily fighting it. Then, coincidentally, a different company, which had previously made me a tempting offer, checked back in. They made an offer to double my TC, which included a big title jump, to fit their pay grades. I wanted to be loyal to my team, so I went to the appropriate exec at my employer. I said I had an unsolicited offer that I had to decide on immediately, but I would stay if we could solve the problem. Was assured exec understood, and we could tackle the problem. I also asked for the company to do right by a couple other employees, while I had the exec's ear and the moment. Existential problem got worse, and couldn't be solved, for political reasons. Everyone was miserable, and I was out the boost to lifestyle and resume decorations.)
The more usual reasons I know not to mess with counter-offers are that: if the employer wasn't treating you fairly before, that's a problem; you might be flagged as disloyal; they might pay to keep you for temporary convenience, but get rid of you when more convenient for them.
(I was at place that had an existential problem, and unhappily fighting it. Then, coincidentally, a different company, which had previously made me a tempting offer, checked back in. They made an offer to double my TC, which included a big title jump, to fit their pay grades. I wanted to be loyal to my team, so I went to the appropriate exec at my employer. I said I had an unsolicited offer that I had to decide on immediately, but I would stay if we could solve the problem. Was assured exec understood, and we could tackle the problem. I also asked for the company to do right by a couple other employees, while I had the exec's ear and the moment. Existential problem got worse, and couldn't be solved, for political reasons. Everyone was miserable, and I was out the boost to lifestyle and resume decorations.)
The more usual reasons I know not to mess with counter-offers are that: if the employer wasn't treating you fairly before, that's a problem; you might be flagged as disloyal; they might pay to keep you for temporary convenience, but get rid of you when more convenient for them.