What type of eye protection should you use with a battery tab welder?
Normally capacitance discharge resistance welders do not create a visible arc- the heating not from an arc and it is confined to a small region where the parts are in contact. 

It is possible to have a small amount of material (i.e. molten metal) ejected from the weld area, particularly if the power level is too high. For this reason, it's recommended to wear safety glasses (an industrial safety procedure would require it). 

His home-brew setup has the potential to create a visible spark, but it should be brief and not very intense. The poor control over pressure means that the risk of ejecting material is greater. 

If it was me, I'd wear ordinary safety goggles.