I'd say yes. The story evolves over the course of the previous two games, and there's really no way to watch/play even the introduction sequence of this game without spoiling some of the major plot reveals of the first.

Things that Shepard discovers in the first and second games are pretty much widely known at the beginning of the third, so the plot twists and turns of those games will be completely lost as people are going to be conversing about them at length even early on.

Some of the returning characters played major roles or were scarred by events in the previous game ("the Virmire survivor" comes to mind as being key), so attempting to talk to them without getting spoiled about the events of previous games is virtually impossible.

That's not to say that starting here will utterly ruin your taste for either of the previous games. I believe Bioware has tried to make this chapter accessible for newcomers, and the plot of the first two is nuanced and complex enough that you'd still gain from going back and playing the first two if you get engrossed in this game's universe.

However, if you're the type that values not being spoiled, starting with ME3 is going to really reveal a ton about the plot of the first two games.

Every Mass Effect 2 save file has a "QuestCompletion" variable that can be one of three values. Gibbed's save editor has dubbed these, "Quest Incomplete", "Lived To Fight Again", and "Went Out in a Blaze of Glory".

There is no "mission complete" save generated by ME2 like there is in ME1. Instead, every save file has this variable in one of the three states. Presumably, only saves where Shepard "Lived to Fight Again" will be viable to import, as Bioware has previously stated dead Commander Shephards won't be importable into ME3. (And obviously, you can't import without beating the game).

(Related: https://gaming.stackexchange.com/a/52493/336)

When importing into Mass Effect 3, you get to pick the actual save to import (i.e., quick, auto, or manual). Your save is always current, so there should be no worry about keeping your save "updated", other than simply saving normally from playing the game.