To "put on the backburner" is to postpone, potentially indefinitely. The closest I can muster to its opposite is to "put back on the table". Is there a better way to to phrase the idea of assuming a line of work/inquiry that had been paused? "Move forward"/"put forward" isn't specific enough because it doesn't imply that the work had been once put off.
"Move to the front burner" is actually an expression too, understood to be the opposite of putting something on the "back burner."
Move to the front of the line or top of the list
Make it a priority
Take up from the table (an actual parliamentary procedure, after something has been "tabled")
Make Job 1 (assign top importance to)
Light a fire under (ignite or rekindle action on something)
Put on the fast track (designate something for immediate action)