An automatic dishwasher detergent dispenser door can become stuck for a few different reasons. Some problems are easy to fix, while some require more work. Repairs are basically the same no matter what brand of automatic dishwasher you have, although specific parts might be required. Fortunately, even if the detergent dispenser needs to be replaced, the problem is not an emergency. The dishwasher will clean dishes fine even with a stuck detergent dispenser.
Move the basket that holds silverware as far from the detergent dispenser as possible, or make sure no tall items are blocking the dispenser. Also, keep other items such as plates and pans away from the dispenser. Anything that winds up touching the dispenser during the wash cycle can cause the dispenser door to stay closed. Run the dishwasher to see if this solves the problem.
Remove the front door panel according to your instruction manual, or find the manufacturer's instructions online, which will probably be in a PDF file. The panel is typically held on by 6 to 8 screws along the edge, which you can see when the door is open. With most models, you won't need to remove the screws at the bottom of the door, which are part of the hinge assembly.
Clean the detergent dispenser thoroughly from the inside of the dishwasher and the exterior side as well. Accumulated detergent and other material can interfere with the spring mechanism or halt the pivot action of the door. Use very hot water and a micro-sized brush. You also can use a mix of hot water and vinegar.
Most dispensers have a bimetallic strip or switch, attached by screws, which triggers the door release. With the front door panel removed, locate the switch, and push on it gently with a screwdriver until the dispenser door opens. You might simply need to adjust the alignment of the switch.
Purchase a new switch if realigning the old one does not fix the problem. Typically, these switches cost under £6. See Resources for a link to different manufacturers where you can buy the part.
Label the two wires connected to the switch so you can correctly reconnect them afterward, and then remove them from the terminals by holding the slip-on connectors --- not the wires --- and pulling firmly. Use a pair of needle-nosed pliers for a better grip. Then remove the switch from the dispenser by removing the screws holding it in place. Connect the wires to the terminals of the new switch, and attach the new switch to the detergent dispenser. Replace the door panel.
Run the dishwasher again to see if the switch replacement has done the trick. If not, decide whether to replace the entire dispenser according to the manufacturer's instructions, or call a repair technician.
Many people don't bother to fix the dispenser. You can leave it open and add dishwasher detergent at the beginning of the cycle as usual. The soap will enter the cycle earlier than it normally would, but the dishes will still be completely cleaned.
Unplug the dishwasher, or turn off the power at the breaker or fuse box before working on it.