AC and DC supplies usually do not get mixed. However, in some circuits where they are used together, there is a chance of mixing. The mixing happens through a common ground, induction across cables which are passing close by.
Another common cause of mixing is while using a dc bias supply to control the transistor current.
Mixing AC and DC can create ripples in the DC supply which can damage electronic cards. The mixing of supplies can also result in blown fuses or other erratic behaviour.
Optocouplers and LC Filters are some of the methods which can help ensure that the AC and DC supplies are kept separate.