1) Borrowing causes higher interest rates and "crowding out" (i.e. discourages private imvestment)...this can occur to such a degree that private investment is reduced so much that it's worse than if the government had not increased spending
2) Timing is a problem. The government intervention usually comes too late, and causes inflation
3) It encourages bigger government (this is only a problem if you are opposed to that politically/philosophically)
4) In the 70s it was revealed that Keynesian polcies do not always reduce unemployment as previously thought. So, instead of accepting a trade off of more inflation for less unemployment, you might get more inflation and more unemployment ("stagflation")
5) Inaccurate demand estimates. Keynesian economics assumes you can know how much demand needs to be increased to fix the economy, however, that can't be accurately predicted.
The biggest problem for most people overall is the tendency of Keynesian policies to cause inflation.
One other potential problem is that, depending on where/how the government increase spending, it may be politically difficult to undo that policy when it's time to cut back on spending after recovery has kicked in. So, if you make a bad spending decision to create a recovery...you might get stuck with it