Will 8 speed shifters work on 6 speed cassette?
Easy answer

No, this won't work with 8 speed indexed shifters. The reason is that one click on your shifter doesn't move your derailleur enough to shift to the next cog on your 6 speed cassette.

Explanation
If we check a table of cable pulls, we see that a 6 speed Shimano cassette has a pitch of 5.5mm. Simply said, this means that the distance between 2 cogs on the cassette is 5.5mm.
An 8 speed cassette on the other hand as a pitch of 4.8mm. An 8 speed shifters will thus be indexed to move the derailleur 4.8mm on each click, 0.7mm less than you need.

Complex answer

An 8 speed shifter won't move your derailleur enough to shift to the next cog, lacking 0.7mm on each click.
Pitch however is described by the shift ratio of your derailleur and the cable pull of your shifter. For Shimano 6, 7, 8 and 9 speed, the shift ratio is 1.7 (moving the shifting cable by 1mm moves your derailleur by 1.7mm).

What you might be looking for, if you want to do some hacking are tricks to manipulate the shift ratio. On the bottom of this webpage, a number of alternative routings are described. In your case, you could try if routing #2 or #3 would work. However, fine tuning the cable pull is not very simple, and might need some trail-and-error and drilling (in case the obtained cable pull is too large).