Learn Italian: C'è un buon presagio qui, lo sento. - There is a good omen here, I feel it.

It could, but here it acts as a pro-clausal pronoun. This means lo replaces the entire preceding statement, rather than just the masculine noun. You are expressing 'I feel that there is a good omen here' rather than just 'I feel the omen'. When replacing a whole idea or fact, lo is always invariable.

If the sentence were plural (Ci sono dei buoni presagi qui), would lo sento change to li sento?

It depends on your exact meaning. If you are referring directly to the omens themselves, you would use li sento to mean 'I feel them'. However, if you are still using the pro-clausal lo to mean 'I feel the fact that they are here', it remains lo sento. This invariable lo is incredibly versatile for referencing entire concepts regardless of the gender or number of the nouns involved.

Why is it buon presagio instead of buono presagio?

The adjective buono drops its final letter before most masculine singular nouns. The rule of thumb is that buon follows the exact same pattern as the indefinite article un. Since you would say un presagio, you must say buon presagio. You only keep the full buono before masculine nouns that take uno, such as buono studente.

Could I use the verb instead of for feeling an omen?