In terms of what might happen on election day, it might take longer to tabulate the votes. If you have more choice that is one of the drawbacks. In, say, STV, it can take longer. The vote-counting process is much more complicated and time-consuming.
Will it increase turnout? Most comparative research seems to indicate that proportional systems provide a bump in turnout of about six to seven percentage points. That may be a reflection of the fact that, again, more currents of opinion are reflected in the resulting legislature, which brings more people to the polls, but it's not a social scientific law. If you want to look at the experience of New Zealand, voter turnout increased in the first elections using the new system, a mixed member proportional, from 1996 on, but afterwards it fell again. The constraints, the factors, the dynamics that are driving down voter turnout extend well beyond the electoral system. I think if you want to get a handle on what's happening, you have to look more and more at what young people are thinking and why they're not coming to the polls in the numbers that we, older, aging boomers, think they ought to be coming to the polls in.
Personally, I do think that if we offer more choice to voters, and I'm just going to editorialize just a second.... I think that so far we've heard a lot about the fears of what might happen with a change: national parties would disintegrate, fringe parties would proliferate. The system has worked well as it is, but you have to ask for whom? I think the one group that consistently has been excluded is young people. I think by offering greater chances to newer parties, non-mainstream parties, you would mobilize more young voters and bring them into the system in a way that the status quo does not.