Does Mass Effect 3 require Origin?

According to Wikipedia, you will need an Origin account for both the single-player and the multiplayer campaigns in ME3.

Customers purchasing PC editions of the game (retail or digital) will be required to install Electronic Arts' content delivery and digital rights management system, Origin. The Origin client (and an Origin account) is required to install, activate and run Mass Effect 3 on a PC for single and multi-player portions of the game.

This Bioware forum post by Chris Priestly gives a lot more detail around Origin and Mass Effect. Specifically, it speaks to your question:

Is constant Origin connection required or is it a single one off authentication when the game is first installed. Is there also a limit to the number of installations available? Mass Effect 3 will require a one time, single authorization for the single player game. There is no limit to the number of installs. Playing Mass Effect 3 Multiplayer will require a constant connection.

Unfortunately, I believe that the answer is yes.

If you read the fine print of this order page, it says (emphasis is mine):

INTERNET CONNECTION, EA/ORIGIN ACCOUNT, ACCEPTANCE OF PRODUCT AND ORIGIN END USER LICENSE AGREEMENTS, INSTALLATION OF THE ORIGIN CLIENT SOFTWARE (WWW.ORIGIN.COM/ABOUT) AND REGISTRATION WITH ENCLOSED SINGLE-USE SERIAL CODE(S) REQUIRED TO PLAY...

Similar to Battlefield 3, currently yes.

All PC versions require Origin. All console versions require an EA/Origin ACCOUNT.