Here is my take on this as a UK speaker (with imagined context added in brackets)...
- You are not to use the elevator (if there is a fire.)
This is a statement from someone in authority to someone junior. It is something that a teacher might say to a pupil. Or, as in my example, a safety notice. You are expected to obey.
- You are not supposed to use the elevator. (But when you are carrying something heavy, it is hard to negotiate the fire-doors on the stairwell.)
This implies there is a rule or current practice that expects you to use the stairs, but the 'supposed' qualification suggests that (a) you are being addressed by an equal, and (b) the unspecified person that expects you to obey is probably someone else. It is likely the next word will be 'but' followed by an exception to the rule. There may be times and conditions when you are not expected to obey.
- You don't have to use the elevator. (If you are only going up one floor it is faster to use the stairs.)
This could be taken as the reverse of the last statement: "You are supposed the elevator". It could be even milder. A hotel may have fancy lifts while the stairs are plain and hidden behind fire-doors, but you are free to use either. There is a suggestion that there is a common practice, and the speaker may be about to provide an exception to this.
None of this is in the grammar; all of this is in the use. Non-UK speakers may have a completely different take on this, as US speakers do with the phrase "I will do it presently".