Many humans need their third molars (wisdom teeth) removed due to tooth impaction. Are humans the only species to suffer from this?
According to Wikipedia An impacted tooth is one that fails to erupt into the dental arch within the expected developmental window. Because impacted teeth do not erupt, they are retained throughout the individual's lifetime unless extracted or exposed surgically.
Now to your interesting question, there are case reports, that yes, animals, other than humans can also have impacted teeth.
Mandibular canine tooth impaction in young dog which was then made to erupt by surgery. Reference
Maxillary canine impaction in a persion cat. Reference
According to this article
Proper growth and development of the oral cavity depends on a series of events that must occur normally and in the proper sequence. Genetic abnormalities or trauma that affects either the developing tissues or the timing of their development can cause abnormalities. Defects that decrease comfort, health, or function require treatment; those that result in only an esthetic problem do not. Common developmental problems include persistent deciduous teeth, unerupted teeth, malformed teeth, malocclusion, and malformed jaws.