Here are some of the records that were broken, will update the answer if I find out more:

 - **Most saves in a playoff game** <sup>[1][1]</sup> : 
85 by [Joonas Korpisalo][2] of CBJ (since shots started being tracked in 55-56 season)

 - **Total combined shots** <sup>[1][1]</sup> : 
151 combined by both teams. CBJ had 63, TBL had 88

 - **Most shots by a team in a game** <sup>[1][1]</sup> : 
88 by TBL (since shots started being tracked in 55-56 season)

 - **Most time on ice** <sup>[1][1]</sup> : 
65:06 by [Seth Jones][3] (since ToI has been tracked in 97-98 season)


  [1]: https://www.sportsnet.ca/hockey/nhl/multiple-records-fall-lightnings-unforgettable-ot-win-vs-blue-jackets/
  [2]: https://en.wikipedia.org/wiki/Joonas_Korpisalo
  [3]: https://en.wikipedia.org/wiki/Seth_Jones_(ice_hockey)