Alex Neil's side moved into second place after the stalemate at the Hawthorns, but they are just one point ahead of Ipswich and Middlesbrough and could be overtaken on Saturday.
Third-placed Ipswich face local rivals Norwich and fourth-placed Middlesbrough host struggling Portsmouth.
Ipswich have three games in hand on Millwall and Middlesbrough have one.
Millwall created the best chances but Josh Coburn volleyed wide from five yards, while Mihailo Ivanovic hit the bar from long-range.
The Lions have 37 points on the road -- the best record in the division -- but it looks likely they will have to seek promotion via the play-offs.
West Brom moved five points clear of the relegation zone.
