I would say that the answer to this depends on how sensitive you are to spoilers. As has been noted, the survival mode is just the multiplayer maps with infinite waves of increasingly difficult enemy forces. There's no plot- or single-player-related spoilers to be had there.
The mission mode draws its maps directly from the single player campaign, but the stories told are all tangential to or from a different perspective from the missions in the single player campaign. In some, the objectives tie directly into story elements, but are provided to you without any context.
Someone will shout "Kidnap X!" or "Save Y!" and then you've got to go and do that thing, but there's no plot context for the mission given in the mission - the story that ties it all together is presented in the single player campaign. At the end of the mission you might know that someone is kidnapped or lived through a kidnapping, but you won't know why that's important or how it ties into the rest of the story.
If you're feeling particularly sensitive, I'd avoid the following missions prior to playing the related single-player campaign mission:
- Milehigh Jack
- Hostage Taker
- Toxic Paradise
All three of these directly relate to plot elements in the campaign, although they are told from different perspectives.
The rest (IMHO) are less plot related and more just rehashes of single player maps with different objectives. My recommendation is to play the co-op with your friend, regardless of whether you've finished the campaign - it's the most fun mode in the game, especially if you've got a good co-op team. However, if you're super sensitive to spoilers, your best bet is to play single player first.