Ice will reduce swelling around an area and can also be effective on pain (numbing action).. Heat will increase the blood flow, which in return brings in more healing elements to the injured section. Just think..Cold-contracting. Heat- expanding.
Probably the more common treatment is to alternated cold & heat, 10-15 minutes each. Twice a day usually. Still some disagreement, from trainers, about this protocol. Another suggestion might be to use a heat rub/lotion/patch during the day. Insure that a steady supply of healing blood will reach the injured area. In workout, this can also work, plus keeping the coverup, as long sleeve sweats, etc.
A tennis elbow can mean many things; nerve damage, rupture/tear, joint/muscle insert problems. Common, and usually most effected, is complete rest for at least 2-3 weeks, depending on age of the person. Could try to work around it, as most people do, depending on the extremes of the potential injury. If still lingering problems, than probably will need and X-ray or such. good luck.