There is also talk of BoJo refusing to resign if he loses a vote of no confidence meaning that the Queen would have to sack him. Crazy stuff.
To the best of my knowledge, he wouldn't have to.
He'd remain party leader, and he remains PM unless and until, he or so some else can command a majority of parliament, and he can't stop them visiting the queen. I guess he could refuse to leave Downing street (in which case, armed police would be on site already to escort him off) or refuse to sit on the other side of the house in which case... I guess they could carry on with business just sitting on different benches.
If no-one can command a majority, then a GE is called, (still with Boris as PM), with the only requirement for timing that it's no less than X days (about 3 weeks), no maximum time frame before holding it. Parliament is dissolved for campaigning from that point. Trying to push that GE into November would be the longest ever, and another constitutional crisis, but as far as I know, there's no means to prevent it - certainly not without taking time in court...
That's why he wants to delay any call for a GE until after the party conferences, which means he could have a 32 day (or less) campaign, which is normal enough.