IQ is positively correlated with sharing economists' views on economic policy Therefore it seems likely people would vote differently and that this would translate into policy changes. I would expect other belief changes would be likely if IQ were increased.

Here is a simple model. Assume you need a certain intelligence to understand a crucial, policy-affecting idea (we can make this a fuzzy border and talk about this in distribution to make it more realistic later). If you are below this level your policy choices will depend on taking up plausible-sounding arguments from others, but it will be uncorrelated to the truth. Left alone such a population will describe some form of random walk with amplification, ending up with a random decision. If you are above the critical level your views will be somewhat correlated with the truth. Since you are affecting others when you engage in political discourse, whether over the breakfast table or on TV, you will have an impact on other people, increasing their chance of agreeing with you. This biases the random walk of public opinion slightly in favour of truth.

From most models of political agreement formation I have seen, even a pretty small minority that get biased in a certain direction can sway a large group that just picks views based on neighbours. This would suggest that increasing the set of people smart enough to get the truth would substantially increase the likelihood of a correct group decision.

My rebuttal to

imagine how much saner the political process would look

is to point at the work of Tullock and Buchanon on Public Choice theory. Basically, the take away is that politicians and bureaucrats respond to incentives. If the voting public were smarter, politicians' behavior would be different during elections, and the politicians would try to make their behavior in office look different. But they would still have an incentive to look like they were addressing problems rather than an incentive to actually solve them. It's much harder than 10 IQ points to align those outcomes.

And bureaucrats and middle managers in government would still face the same incentives about obfuscating results, multiplying staffing and budgets, and ensuring that projects and bureaucracies have staying power.

It's highly non-obvious that it would have significant effects

The effects may well be profound if sufficiently increased intelligence will produce changes in an individual's values and goal system, as I suspect it might.

At the risk of "argument from fictional evidence", I would like to bring up Poul Anderson's Brain Wave, an exploration of this idea (among others.)