The premise of the question is skewed. Words containing ss and words containing ß usually have nothing to do with each other etymologically. Similarly to the English homophones knight and night where it would be pointless to ask ‘do the meanings of other words beginning in n change if I add a k?’
The distinction between ß and ss is a phonemic one that stems from the distinction of short and long vowels (and that of voiced and unvoiced s). It only originated this way because that is how our script originated from its Latin original. Modern Latin usually distinguishes between long and short vowels by adding a macron onto the former (and maybe a brevis onto the latter) but that distinction was not present in classical Latin. German handwriting can be dated back to Charlemagne and even earlier making it a rather old script that follows many more traditional rules rather than logical ones, not unlike English or French.
Other languages whose writing systems were formalised much later, such as Czech (around 1400 by Jan Hus) and Finnish (around 1500 by Mikael Agricola) chose more logical distinctions between long and short vowels that did not affect the spelling of following consonants — acute accents in Czech and vowel doubling in Finnish. German by then had gotten used to the doubled consonants following short vowels that the system was kept.
So of course, you can go around in different languages, asking, e.g. ‘Are there Czech words whose meaning dramatically changes when a vowel looses its accent?’ or ‘Are there Finnish words whose meaning drastically changes, if a single vowel is written instead of a double one?’* But I really don’t think either of those questions is in any way good. In fact, they would be better described as:
Can I change a phoneme of a word to a different one to change the meaning?
And the obvious answer is of course!
There is one counterexample to the general rule where two words obviously derived from the same precursor: Floß and floss (although one is capitalised, being a noun, and the other is not). Both somehow derive from the verb fließen (to flow).
There is one single case I know of that breaks the relationship of ‘either ß or ss is correct, but never both’: The word Geschoss when meaning floor or storey. The ground floor can be both Erdgeschoss and Erdgeschoß. But this is because there are two possible pronunciations of the word, one with a short o, one with a long o.
Also note that this entire answer covers normal texts in Germany, Austria, Luxemburg and places in which a German-speaking minority has close ties to one of these. In Switzerland and Liechtenstein, ß doesn’t exist and ss is always written where the others distinguish between ss and ß. Also, spelling rules dictate for any variety of German that when capitalising (this includes small caps), every ß is replaced by SS.
*: In fact, Finnish does have a distinction which is closer to relevant: a verb ending in a long vowel is third person singular present (e.g. tulee he/she/it comes/is coming) and a short vowel is second person imperative (e.g. tule come!). But in spoken Finnish, these forms have merged.