Yes getting your ears syringed can cause temporary tinnitus which should go away within a day or so of having your ears syringed.
Other complications of syringing your ears are ear canal pain or other discomfort, temporary dizziness, possible damage to the external canal and even sometimes vertigo.
In some rare cases ear syringing can even damage your ears so only have your ears syringed if you absolutely need too and have tried cleaning ear wax out at home.