I tried binding a key to +speed, the way it is in HL2, but that didn't do anything. I also messed around with some cvars like cl_forwardspeed, gave it all kinds of wacky values, and didn't notice any effect. (Tried with cheats on and cheats off.)
Does anyone have any ideas how I might get this to work?

If you are building maps you can add a suit to achieve this, but on pre-existing maps you may have to cheat to be able to sprint.