If you start dating someone with a pet and you are not used to it, or he is not used to you then it can be hard. There are some people who do not grow up with pets so they truly do not know how to act around them. You may actually have to have a talk with them about it. If your pet does not like your new friend then that can be difficult to adjust to as well. Dogs are especially territorial and they might pee on his things or show their teeth. Hopefully you new squeeze is patient, or you may have to get between them.
They could become jealous of the pet if you spend a lot of time with it. When you are single you have more time to take care of a pet, but when you are in a relationship and want to do some things then you may have to explain that the pet needs to be taken care of first.
Another way that a pet can come between you and your significant other is if they have allergies. They may not be able to stand staying in the same house with your pet, and then you might be forced to make a really hard decision. For a fairly new relationship if you end up having to get rid of the pet, then this can be devastating and the person could harbor some resentment meaning that the relationship is then doomed from that point on.