Learn German: Ich wasche mein Gesicht mit warmem Wasser.

The preposition mit always requires the dative case. Here, the neuter noun Wasser is used without an article like dem or einem (this is called the "zero article"). Because there is no article to show the case, the adjective has to do it using the strong declension. The strong ending for dative neuter is -em, making it warmem Wasser.

Does mit always force the following words into the dative case?

Yes! mit is one of the strictly dative prepositions in German. Whenever you use mit, the noun phrase that follows it must take dative endings.

I often hear Germans say Ich wasche mir das Gesicht. Is that different from this sentence?

While Ich wasche mein Gesicht is grammatically correct and translates word-for-word from English, native German speakers often use a reflexive structure for body parts. They will frequently say Ich wasche mir das Gesicht, which literally means "I wash the face to myself." Both are perfectly fine for you to use.

How can I figure out the gender of Gesicht from this sentence?