My device is recently unrooted, so I decided to update the software of my Samsung Galaxy Tab 3V (SM-T116BU) but this error message (The operating system on your device is modified in an unauthorized way...) always shows up each time I tried to update the software even though I have unrooted my device. Is there a solution for this problem?
did you try with "Smart Switch" program from your PC, and download the update from there? As I saw in the video, you have to download the program, and check the latest update for your phone when connecting via USB. You can check this video, I think you can solve your problem by this way.
Regarding the issue of the tripped dm-verity / etc. counters, I would suggest that you refer to or post up at www.xda-developers.com for further help on that. Assuming there is a current method to reset the counters, that should allow for OTA updates to work again.
For OTA updates you aren't able to install in the mean time, I believe you can download them directly from Sammobile (reputable source). This page is for the model tablet you disclosed, please choose accordingly for your carrier. If you need further assistance, let us know! Good luck.