Just to inform I have the same login problem. I can confirm the issue.
I am installing a brand new forum, lucky enough to delete and reinstall everything from scratch.
I am using phpBB3 3.2.1 with php 7.1 and right after install I can go to ACP. I can setup foruns, permissions, options...
But once I log out, I cannot log in again. Similar to your issue, I have no error message and back to board index... But logged out!
The solution for me was to use Internet Explorer to login. Google Chrome fails all the time.
If some expert request a test, some information, please let me know. I have no idea why this is not working with Chrome but I have a brand new site for testing purposes. Also, please note I have a PC and a notebook, another PC at university and also come friens with notebooks... The issue persists with different Chrome versions, I was trying with different versions and issue persist with Chrome.
Many customers will for sure use Chrome so It´s important to check what is going on.
Good news: I had no problems related with php version 7.1 and default theme.