Learn French: Nous avons payé les impôts hier.

This sentence uses the passé composé (past tense). In French, you cannot just use the past participle payé (paid) on its own. You must use it with a helper verb, which here is avons (have), to mean "we paid" or "we have paid".

How is les impôts pronounced? Do I pronounce the s in les?

Yes, you do! Because impôts begins with a vowel sound, you must make a liaison. The s at the end of les is pronounced like a "z" and connects to the next word, sounding like "lay-zim-po". The s at the end of impôts remains silent.

Since this lesson is about polite requests, how would I change this to say "We would like to pay the taxes"?

You would use the conditionnel form of aimer (to like) or vouloir (to want). You would drop the past tense avons payé and instead say Nous aimerions payer les impôts or Nous voudrions payer les impôts.

Where else can I put the word hier in this sentence?

Time words like hier (yesterday) usually go at the very beginning or the very end of a sentence. So, you could also say Hier, nous avons payé les impôts. You should not put it in the middle of the verb structure.