The Channel Islands played a minimal role in the D-Day invasion. This was thanks to their geographical position; they are on the western side of the Cotentin Peninsula, while the invasion beaches were positioned to the east of it. With the peninsula in the way, the German forces based on the islands could not affect the landings, letting the Allies could ignore them and focus on the beachhead.
During the run-up to the invasion, the Channel Islands saw fairly frequent attacks by Allied aircraft. These raids targeted fortifications and radar sites on the islands. They also hit shipping around the islands. These raids formed part of a general Allied policy of bombing targets along the northern coast of France. By spreading out these raids, the Germans were kept guessing as to where the invasion was going to fall. In a similar context, the islands were part of the targets for Operation Hardtack, a series of commando raids on the French coast. Carried out in late 1943 and early 1944, these raids were intended to reconnoitre beaches in possible invasion areas and gather intelligence on the German defenders. By spreading out the location of these raids, the Allies again helped to conceal the true location of the landings, as well as gathering wider intelligence. Hardtack raids hit two Channel Islands; Hardtack 7 on Sark and Hardtack 28 on Jersey. Neither raid was a success. On Jersey, the commandos gathered a small amount of information on German deployments before running into a minefield on their return to their boat, while on Sark, they hit a minefield shortly after landing and were forced to withdraw. Three commandos died during the two operations, two on Sark and one on Jersey.
During the D-Day landings themselves, the islands played a very minor part. The aircraft carrying American paratroopers to their drop zones behind Utah Beach followed a flight path that came down the eastern side of the Cotentin, before turning east towards the drop zones. This flight path steered clear of the Channel Islands, to avoid concentrations of anti-aircraft guns on them. Even so, the Channel Islands made a useful navigational marker for the pilots. Navigating at night over water is quite hard; having a set of islands to mark the turning point greatly simplified the role. Beyond that, though, the islands were untouched.
The German defences of the islands did, however, play a part in the days following the initial landings. As I have described in this comment, the Germans sortied a number of U-boats to interfere with shipping off the Allied beachhead. While most of these were caught before entering the Channel, a number of the boats fitted with 'snorkels' (which allowed them to charge their batteries while submerged) did manage to make it close to the invasion beaches. However, they found operating near the invasion beaches all but impossible. Subs that attempted to use their snorkels in the area were soon located and attacked by Allied ships and aircraft - yet they only had enough battery power to stay underwater and active for less than 24 hours. The Channel Islands, with their heavy coastal guns and AA defences, provided a safe haven for U-boats to surface and charge their batteries. Several boats would put into St Peter Port on Guernsey on the 14th-19th June 1944. Most of these boats saw limited success in the Channel, with one, U-269, being sunk by HMS Bickerton. U-984, though, did successfully damage a British frigate and four Liberty Ships from Convoy EMC-17 in the days after leaving St Peter Port.
In August, with American troops marching up the eastern side of the Cotentin Peninsula towards the key port of Cherbourg, some of the coastal defences on the islands went into action against them. The German Navy's Batterie Blucher, based on Alderney, was most responsible for this. Blucher's fire drew a major response, with the British battleship Rodney being detailed to shell the battery. Firing from a position near Cherbourg, she fired seventy five 16in shells, with spotting provided by an RAF Spitfire above the island. A contemporary newsreel of the bombardment can be seen here, with some unaired B-roll here. The shelling was impressive, but largely ineffective. Two German soldiers were killed, and only one of the battery's four guns suffered serious damage; the battery was back in complete action by November. Fortunately, the same could be said of the battery's bombardments, with only one American soldier being killed by its fire.