None as long as weight loss is the plan.

The "issue" here is that the said girlfriend probably didn't ask for anything and that offering the DVD can be seen as an innuendo for "you're fat, move your ass".

I mean, OP clearly said his girlfriend -is- overweight, so maybe a tiny bit of innuendo here is acceptable, when it comes with a free gift to actually help do something about it. It's merely acknowledging the truth. If OP's girlfriend was already at a healthy weight....then I could see a major issue here. And I say this as a person who is also quite fat, and I would not take offense if someone offered me free workout DVDs(except for the fact DVD technology is pretty out-dated and I don't even have a DVD player)