I am 50/50 with puffer and anemones. My porcupine was fine with my bubble tip and aptasia. My anthron cleaned my rock of everything, bubble tip, aptasia, sponges, caluperia and whatever else he came across.
Looking specifically at a web bur or spiny box puffer. I know the eat inverts, a anemone is a invert, so I am asking for that reason. I figure inverts they eat as snails, but you never know.
When it comes to fish eating livestock I personally consider anemones to be grouped with corals. I don't know why, I just figure if they'll munch on an acan, why not a nem too. Sure them nem has a defence mechanism, but some fish are too stubborn to give up that easily!
I would only trust a puffer with nems if it came from another one if my own tanks, or a trustworthy friends tank who has kept that particular individual before with no coral/nem eating issues.
Just my 0.02