Oliver Hart said that financing deficits by printing money can lead to hyperinflation "once the economy is close to full capacity". What does he mean?

Full capacity means utilization of all resources in economy. That is all labor, capital and other resources are already used at their best uses. Economy will not always work at its full potential for various reasons. For example, due to fall in aggregate demand during recession, which can cause unemployment to be above natural rate.

Monetary policy or fiscal policy can stimulate economy but only to the point economy reaches full capacity (i.e. all resources are put to their best uses). After that point as Hart mentions monetary stimulus will likely lead just to inflation or potentially hyperinflation.