This would have no effect at all.  It's only seems like a good idea for people with bad math skills or the people that wouldn't delete part of their holding which i assume would be everyone unless they were stupid.  If you have 5% less total coins and the price therefore goes up 5% you are still in the same position.
If i owned 21mil bitcoins and they were worth $1,  $21M.   I then burned 1 million BTC so now effectively they are worth $1.05, x 20M  still $21M.  Market cap is what you want to see rising.