Learn Russian: Я фотографирую дом.

Because you are doing the action to an outside object (the house). The endings -ся and -сь are used when an action reflects back onto the subject (like photographing yourself) or for mutual actions. Since this sentence has a direct object (дом), the normal, non-reflexive verb is used.

Why does the verb end in -ую? Isn't the dictionary form фотографировать?

Yes! For many verbs that end in -овать, the -ова- part drops out and changes to -у- in the present tense. That is why it becomes я фотографирую instead of 'я фотографироваю'.

Why doesn't the word дом change its ending since it is the direct object?

Дом is indeed the direct object and is in the accusative case here. However, for inanimate masculine nouns (objects and places, not people or animals), the accusative form looks exactly like the nominative (dictionary) form. The ending would only change if the masculine noun were alive.

How would I say this in the past tense?

You would say я фотографировал дом (if you are male) or я фотографировала дом (if you are female). Notice that in the past tense, the -ова- part of the verb comes back!