I have never seen the program, but it seems a fairly straigt forward joke to me:
A dog is for life and not just for Christmas;
Children often want animals for Christmas, but get bored with them by New Years, because they have the attention span of, well, children. So the first part is often said to make it clear to the child, that if you actually do give them a dog, they will have to care for it for a very long time, take them for a walk in any weather and even if they want to play an online game with their friends.
so do be careful at the office party.
Around Christmas time their are usually office parties to celebrate the season, often accompanied by the consummation of... alcoholic beverages, like eggnog. Sometimes in this inebriated state people get promiscuous, which could result in them also getting a christmas 'gift' they'll have to care for life for.
Basically the joke is about the alleged fact that around christmas time the probability of cheating goes up and if you're too drunk - not careful that is - to think about contraceptives, you may end up pregnant and be burdened with a life long commitment you didn't want nor had thought through.