I'm in the same boat. Can't find any recent definitive exploration of LR and TM.
I have used LR since the beginning, and there were lots of websites at the time warning not to back up the LR catalog using TM.
I am now setting up my new iMac from scratch (rather than using Migration Assistant) because I'm keen to revisit issues like this, rather than continuing to rely on decisions I made as many as eight or so years ago when I first moved to Macs.
The Lightroom Help says:
Running Mac OS X Time Machine™ backup or restore operations while Lightroom is in use is not recommended.
Unlike modern backup systems, Time Machine doesn't take consistent point-in-time snapshots of the entire disk. So it's possible that a Time Machine backup of your catalog files will be inconsistent, from different points of time as the catalog is being modified by LR, and thus not able to be restored properly.
If you use Time Machine, you should make sure that you exit Lightroom for at least a couple hours every day, to give the hourly Time Machine backups a chance to back up the catalog when LR isn't running. Even better is to let Lightroom make its own periodic catalog backups onto your hard drive, and then let Time Machine back up those LR backups. (Those LR backups can be safely backed up by Time Machine at any time, regardless of whether LR is running.)
Workstation/end user backup applications do not always handle open files predictably/gracefully ("modern" enterprise backup systems use custom application-specific agents to address this issue).
Expanding on what John R. Ellis recommends, the best way to ensure you get a reliable backup of your OS X computer and all application data is:
Reboot and open nothing (I even hold the Shift key down during bootup to bypass login items)
Start your Time Machine backup
Don't do anything else until the backup completes (or only open applications that contain data you could live without it it doesn't restore properly- most OS X stock apps are fine to run... I would not open Mail.app though).
In addition to Time Machine incremental backups, I've used Mike Bombich's Carbon Copy Cloner (CCC) for many years to obtain a full backup- occasionally when saving a yearly snapshot of my system and before I perform a major operating system or significant application upgrade. I've had to revert to a full restore of my Mac multiple times in the last 10+ years due to a botched operating system install or an application upgrade gone ugly.
"modern" enterprise backup systems use custom application-specific agents to address this issue
I was referring to backup systems that create true point-in-time snapshots of the file system, such as those available for Linux or Windows.