Model # and manufacturer?
Measure the input resistance when off. What is it? Maybe a relay disconnects the input.
Tek DPO 2024
50 M\$\Omega\$ @ BNC when off, I guess that's pretty safe.
1M\$\Omega\$ @ BNC when on
I believe the MSO/DPO2000 Series is entirely relay free (I’ve never heard one and don’t recall seeing any on the teardown pictures I’ve seen), so the front end isn’t entirely disconnected on shutdown. So, at 50 MOhm and, with a 10x probe, you should be good to the maximum input ratings of the scope, off or on.
That said, you should get in the habit of disconnecting the probes from the circuit under test when you’re done for the day. Leaving things connected overnight can cause accidents and frustration the next day (accidentally cross connecting the grounds of two loads that need to be isolated, having the probe slip off and spending 30 minutes chasing it down as a scope issue, etc.)
I’ve learned it’s almost always better to start fresh each day. (This rule only applies to actual scope and multimeter probes; obviously something like a logic pod with minigrabbers hooked to the tiny pins of a SMD IC would stay attached overnight, though I would do a wiggle check and make sure they’re all still connected the next day.)