Learn Polish: Przegrałem zakład, ale oby o tym zapomniała. - I lost the bet, but hopefully she forgot about it.

No, the particle oby always requires the past tense form of the verb (specifically, the L-participle), much like when you form the conditional mood. Whether your wish is about the past, present, or future, you must use this past tense structure. So, both "hopefully she forgot" and "hopefully she will forget" are expressed as oby zapomniała.

Why do we use o tym instead of just using to for "it"?

The verb zapomnieć (to forget) is most commonly used with the preposition o (about), which forces the following noun into the locative case. The locative form of to is tym. While you can sometimes use zapomnieć with a direct object in the accusative case (zapomnieć to), zapomnieć o tym (to forget about it) is the standard and most natural way to phrase this.

The English translation says "I lost" and "she forgot", but there are no pronouns in the Polish sentence. How do we know who is doing what?

In the Polish past tense, the verb endings tell you both the person and the gender. The -łem ending on przegrałem means "I (a male) did it." The -ła ending on zapomniała means "she did it." Because the verbs give us all this information, adding the pronouns ja (I) and ona (she) would sound overly repetitive here.