No, I never had the problem with my 3870 Mac & PC original in my 2,66 2006 Mac Pro. I fitted the card after I upgraded to 10.5.7. Prior to this I had another 3870 in a PC and did not like the noise with Windows XP32 and XP64. So I bought an AKASA Neo Vortexx (19€) with dual heat pipes and hot air extraction via 2nd slot. I fitted that cooler before I seriously did anything with my 3870 Mac&PC. The card is inaudible in OS X and Vista Business 64-Bit. The only issue I had when fitting the cooler was the fan connector. The card has a two wire (black and red) cable and the AKASA fan had a three wire (black, red, yellow) cable with a much bigger connector. I made an adaptor by cutting the connector from the old cooler and fitted that to a three pin fan extension cable. The AKASA cooler fits over the heat sink for the voltage regulators, so I left that on. I also had to re route the fan cable around one of the capacitors but that was child's play.
The voltage regulators are most likely to cause problems on this card. A compounding problem may be the opposite blowing direction of the card fan to the PCI-e fan of the Mac Pro. The air stagnates and the voltage regulators may age prematurely. For trial I would remover all slot covers from the back of the PCI-e bay. If the problem is reduced you need to do something for cooling the regulators, although they may have taken a lethal hit already.