I think it will depend on how much of the money makes it to consumers and how the Fed shuts it back down.
During the 2008 crisis, the Fed was fairly confident inflation wouldn’t be an issue because the unemployment rate was moderate. The line of thinking goes that one of the drivers of inflation is wages, wages don’t go up until there’s a shortage of workers, and the indicator that there’s a shortage of workers is a low unemployment rate.
The aid currently being developed is based on MASSIVE unemployment. Applying the 2008 situation to today, inflation isn’t going to be driven by high wages. (There have got to be other drivers of inflation I don’t know about, and those may override the unemployment aspect.)
So then it becomes an issue of how they shut it down when the economy picks back up. If they can manage to withdraw the aid at about the same rate the economy picks back up, then there shouldn’t be too much extra money in the system to drive inflation. On the other hand, if they’re slow on withdrawing the money, then I would expect some inflation.
On second thought, they’re bound to screw up the withdrawal. So never mind. Expect inflation. How much will depend on the fortitude of the Fed.