Learn Finnish: Suojaa lasta, koska tilanne on paha. - Protect the child, because the situation is bad.

suojata is a Type 4 verb. To form the 'you' (singular) command, you remove the -ta ending from the dictionary form to get the base suoja-, and then add a vowel that matches the last vowel. This gives us suojaa (Protect!).

Usually objects in command sentences don't have an object ending. Why is lapsi in the partitive form lasta here?

You are right that complete objects in commands often drop their object ending. However, the verb suojata (to protect) is a partitive verb. It always forces its object into the partitive case, because protecting is seen as an ongoing, continuous action.

As a quick reminder, why does lapsi look so different as lasta?

lapsi (child) is an older Finnish word with a unique stem change. To make the partitive form, the -psi part changes to an -s- before we add the partitive ending -ta. This makes it lasta.

Is the word tilanne (situation) in a special case here because it ends in -e?

No, tilanne is simply in its basic dictionary form. Finnish has a whole group of words that naturally end in -e, like huone (room) or osoite (address).