Bits of meat sticking to the vegan burger would be a big no-no for me and the vegans that I know!
I'm not personally bothered by using the same tools to handle both meat and vegan options, as long as I can't taste it, if we're talking about backyard barbecuing with friends. But I expect more from a company.
I had an unpleasant experience a couple of years ago at Subway. I ordered a vegan sub, and the person behind the counter asked if I wanted them to switch their plastic gloves. "No need," I said, because I thought it was a moral question. But when I started eating the sub I could smell meat along the whole bread as I ate it. It was not pleasant when it was two decades since I last had meat, but I was drunk and hungry and finished it anyway.