In many cases, the 3 verbs +”oter” are synonyms. For example, from a list of names, you may indifferently “Oter, retirer, enlever ou supprimer un nom de la liste”, i.e. remove one name from the list.

There are nuances in some of these verbs usage:

• “Retirer” : To put something aside. [translation: to withdraw]
• “Enlever” : To take away something from somebody; also refers to a substraction. [closest translation: to remove]
• “Supprimer” : To fully delete something (or someone). [translation: to delete]

retirer, enlever et ôter are very much interchangeable, the later being a bit old and some expressions commanding more one than the other. They all imply that what you remove still exists but is put somewhere else.

supprimer implies the destruction of the element removed.

For example:

retire tes vêtements, enlève tes vêtements, ôte tes vêtements are equivalent and can be used in real life…

supprime tes vêtements can only be used when erasing clothes with photoshop on a selfie, or when making a final decision to become a full time nudist.

enlever is preferred when you lift the thing you remove: enlever une voiture (tow a car away) because in France, the police use trucks with a crane for this.

retirer is preferred when you pull something back, for example when playing bridge, if you take a card back into your hand after putting it down. You can also use reprendre for this particular case. It can be used figuratively as in je retire ce que j'ai dit (I take that back)

Those 3 verbs have roughly the same meaning. There are some nuances though :

• "Retirer" is quite well translated to ‘Withdraw’ in several meanings : retirer de l’argent (to withdraw cash), retirer sa plainte (to withdraw a lawsuit), se retirer d’un accord (to withdraw from an agreement).
• "Enlever": ‘to remove’ or ‘to take something away’. Also, ‘to kidnap’. More rarely ‘to collect’ as in garbage collection (enlèvement des ordures)
• "Supprimer" : ‘to erase’, ‘to delete’ or ‘to make something disappear’. Does not imply destruction as it does not generally refer to a physical objects, but mainly to an intellectual creation : a part of a document, of a speech (supprimer un chapitre, un paragraphe, une ligne, un mot), of a song (supprimer un couplet), or a file on a computer (un fichier).