Often when we get questions about whether to use a participial or an infinitive clause, the answer depends on the requirements of the particular word that governs the clause (eg like in "I like swimming in the pool" vs "I like to swim in the pool").
But in this case, there is no word governing the clause in question: they are adjuncts, rather than complements of anything.
When an infinitive clause is an adjunct, it is usually a purpose clause: so to stand up or the most hated man in the world is explaining the purpose of what it follows: giving a 20-minute speech. Your sentence could be paraphrased as Giving a 20-minute speech for the purpose of standing up for ...
A participial clause, on the other hand, qualifies a noun phrase, but does not express any specific relationship to that clause. A paraphrase might be a 20-minute speech which stands up for ... (or maybe which stood up for).
In this context, there is little difference in meaning that I can find; but in other cases there might be.
If in fact the speaker was not intending to stand up for the most hated man, and didn't realise that their audience would take their speech in that way, the it might be a speech standing up for but would certainly not be a speech to stand up for.