Is there any valid rule discouraging the use of a certain word to start a sentence?

Well, with certain words it's simply impossible to start a grammatical sentence: one such word that comes to mind is "ago". It always comes after other words (e.g. "one hour ago"), never at the beginning of a sentence or clause.

[Before someone points it out: note the use-mention distinction. A sentence like

'Ago' is a word you cannot start a sentence with.

starts with the word "'ago'" and not with the word "ago".]

But if your question "Is there any valid rule discouraging the use of a certain word to start a sentence?" (emphasis mine) implicitly restricts attention to words that can grammatically start sentences, then it's not clear what it would take for a rule that discourages something grammatical to be "valid". Certainly there exist people who disapprove of certain words starting sentences for their own idiosyncratic reasons, such as the "but" I started the previous sentence with. Are these "rules" valid? I wouldn't consider them valid, but I don't know what valid means to you. :-)

In school we were taught to "never begin a sentence with 'and', 'but', or 'so'". But I do anyway. And so do a lot of other people.