The first problem is that you need to check the value of $minute every time you change it, the second problem is that, because you have two <<if>> statements, you're checking the value of $minute again after you've just changed the value of $minute, so the second <<if>> probably won't get triggered. The correct way to do what you're asking for would be like this:
<<set $minute to $minute+70>>
<<if ($minute gte 60)>>
<<set $time to $time-1>>
<<set $minute to $minute-60>>
<</if>>
However, if you added 130 minutes, then that would only remove 60 of it. That could be fixed like this:
<<set $minute to $minute+70>>
<<for $minute gte 60>>
<<set $time to $time-1>>
<<set $minute to $minute-60>>
<</for>>
The <<for>> macro will keep looping until $minute is less than 60.
That said, it's a bit weird that you're adding to minutes and subtracting from time. You might make things a bit simpler by only subtracting minutes and using a JavaScript Date object, like this:
<<set $Time = new Date('1/3/1970 00:00:00')>>
Time remaining: 24:00:00
<<set $Time = new Date($Time.setMinutes($Time.getMinutes() - 15))>>(15 minutes later...)
Time remaining: <<= $Time.toLocaleString("en-US", { hour12: false, hour: "2-digit", minute: "2-digit", second: "2-digit" } )>>
This way you just have to subtract the minutes, like how it's done on the third line above, and the Date object will handle the rest. Then, when $Time.getDate() == 1 you know that more than 24 hours have passed. (Note: <<= "xyz">> is just a shorter way of doing <<print "xyz">>.)
If you want to show "24:00:00" instead of "00:00:00" you'll have to fake it like I did above.
Then, to make things easier, you can turn those commonly used lines into widgets. Just create a passage, give it "widget" and "nobr" tags, and put this in it:
<<widget "PassTime">>
<<set $Time = new Date($Time.setMinutes($Time.getMinutes() - $args[0]))>>
<<if $Time.getDate() == 1>>
<<goto "TimeOver">>
<</if>>
<</widget>>
<<widget "ShowTime">>
Time remaining: <<= $Time.toLocaleString("en-US", { hour12: false, hour: "2-digit", minute: "2-digit", second: "2-digit" } )>>
<</widget>>
Now you can make 15 minutes pass and show the time like this:
<<PassTime 15>>(15 minutes later...)
<<ShowTime>>
That will also send it to the "TimeOver" passage if time runs out using the <<goto>> macro. (NOTE: Other code called in the current passage will still get executed after the <<goto>> is called and before going to the passage you told it to go to, so use <<goto>> macros with caution.)
Hope that helps! :-)