Round 1 : 5 Mile Race
Lee Weights off no gates
Round 2: 5 Mile Race
Lee 5 open gates
Fight in the chunin exam arena
Lee is drunk no weights and can open 5 gates
Easy win for the Raikage.
If Lee could somehow open all gates though (Possibly in Part 2) then his power should surpass that of the Raikage's and in a 5 mile race I would assume it could last long enough to give him the win.
Edit: I missed the 5 gate limit, yeah Lee loses.