I have submitted a bug report for this.
I used a gold seal hireling to help me with a quest. When I was done, it had 53 min remaining. I placed it in my shared bank to make room in my inventory. I ran some quests, and logged out for the night.
Today I was poking around in my inventory while my wife was getting ready to go out shopping (joy...). I noticed the gold seal hireling reported 13 hours 9 min remaining (paused). Thinking that was odd, I moved it to my inventory. It reported 53 min, and the timer began to count down (clock face on icon, 53 min NO (paused). I then moved it back to my Shared Bank, where it lost the clock face, but still didnt get a (paused).
I waited until it said 52 min remaining to confirm the clock was counting down. Then I logged out and back in. The timer was reset to 53 min, but was still not paused. I then typed up a bug report, and checked on the hireling. 43 min remains. Logging out and in a second time reset it back to 53 min.
I have not tested to see what happens if I use the hireling in this state.