It may surprise you to know that there aren't many non-standard effects on his voice. Here's a quote from mixonline:
“We changed mic pre and compression settings to bring out the depth of David's vocals. We overdrove the compressor. We used a Distressor on the vocals occasionally when we tracked, but it was not always engaged. The vocal mic pre was an Avalon 737, and I used a Neumann tube as a vocal mic.”
I was also reading that David really tries to do his vocals "live," meaning no autotune to fix mistakes, and not much comping together.
So three effects? Compression, overdrive, and.... probably EQ, but they don't mention it.
I did see an article saying that they sometimes will record doubles for his voice, pan it hard, and kind of bury it in the mix. This strengthens his voice without that floating feeling that hardpanned doubled vocals can get.