The cypher system is extremely simple, which is one of it's great strengths, and I think the system by itself can stand up just fine without cyphers. After all, ultimately they are just items, and there is an upper limit on how many you can carry anyway, so they aren't required to be able to play a normal game.
However, virtually any of the existing written adventures would be very dfficult without them, nor would the setting really feel the same. All the existing content is written with the assumption that you are obtaining and using cyphers. With cyphers being so wildly powerful (if only for a limited time) this makes it a safe assumption that players can handle situations that would normally be beyond their capabilities, making for fun, tense encounters that are saved by an interesting or novel use of a cypher. Without them, you are either sending players into a meat grinder or you need to scale things back.
So to answer your question, I feel that yes, you can use the Cypher system without cyphers, but you can't use any of the existing content without modifying it. I've observed this effect directly in my own groups, where the players, being new to Numenera, are treating cyphers very much like D&D magical weapons and hoarding them rather than using them as disposable items. Because they are hoarding them, they are effectively out of play and it's changed the game's balance considerably, making encounters as written in adventures such as "The Seed Ship" much, much more difficult for them. I would have lost at least 2 players in their second session had they not decided to start using their cyphers at the last minute.
Without cyphers, the game plays a lot like any other rules light, narrative focused game, such as FATE for example. What cyphers provide beyond their actual abilities is an injection of weird for the players and the GM to react to. They are more than just items of power, they establish and reinforce the tone of the narrative and it's focus on weirdness. This isn't to say that you can't bring that in by other means, but without cyphers that requires more conscious effort on the part of the GM rather than relying on the randomness of player choice.