Utter baloney. How can a building absorb energy from a battery?

You just hadn't recharged your cell phone recently.

A couple of thoughts come to mind:

(1) Cellphones will tend to raise or lower their transmit power when
"talking" to the cell tower, based on signal-strength feedback
from the cell-tower controller. If the hospital building is well
shielded but not entirely radio-opaque, some cellphones might
increase their TX power to stay connected to their network, and
this could run the battery down faster than if they were out in
"clear air".

(2) The nurse might have been making up a story ("save your battery")
instead of saying something like "Please shut down your cellphone,
since we have sensitive electronic equipment in some of the rooms
around here and your phone might interfere with them." She may
have learned that people are more likely to comply with a "turn
the cellphone all the way off" request, if it's excused by a
statement that seems to be in the person's own best interest
("save your battery from being killed").