What it means
To get lumbered means you got stuck with the chore, problem, or boring responsibility nobody else wanted, and now it's somehow your baby. It's very British and usually said with a bit of a groan, because you've been landed with the rubbish end of the deal and there's not much wriggle room.
Usage examples
"I got lumbered with organizing the whole office party again, nobody else would so much as lift a finger."
"I got lumbered with the year-end report because nobody else volunteered to stay late on Friday."
"Don't pick up the phone in the next ten minutes or you'll get lumbered with the new client's tantrum."
"I got lumbered with locking up again because everyone else vanished at five on the dot."
"She knew if she stayed quiet for too long, she'd get lumbered with doing the rota for the whole month."
Where it comes from
It comes from an older English use of lumber for unwanted bulky junk or clutter. That sense grew out of Lombard pawnbrokers' shops, where unclaimed goods could pile up. So if you're lumbered with something, you're stuck carrying awkward dead weight, whether that's actual stuff or a thankless task.