Next to which shrine? The door to the dungeon at the base of the tower (by the lake shrine) is opened after you enter the armoury near the top of the tower, find something there and use it (or turn in the related quest and get another quest, but using it yourself and then turning in the quest is better). The door to Lovis' room is on the level directly above the citadel teleporter shrine. All 3 doors are just called 'door', though, at least after they have been opened for the first time.
If you explored a bit north of Lovis' tower, the entrance to Maxos temple is blocked off by a locked door, which will be opened after you have talked to Lovis. If you made it into Lovis' room but couldn't find a way to proceed, jump over the caved in passage left of the entrance as you enter, and check the other side...
If you have not gotten there yet, make your way up the tower. Shortly after the gate opened by the 4 dragon statues the path branches; keep left to spiral up to the armouy, or right to get down to Lovis' chamber.
