Anytime you remortgage, regardless of house price movements at the time, the lender will value your house before deciding whether or not they will lend to you.
This can effect you when remortgaging for several reasons.
Worst case senario, you mortgage is of a greater value than your house. This is a situation known as negative equity. This will mean you can't go to a new lender as they will not lend you an amount greater than the value of your house (not anymore anyway - see Northern Rock). However in this case you should still be able to stay with your current lender who will hopefully offer you a new deal and worst case leave you on their Standard Variable Rate.
Other problems with your house falling in value are if it takes you beyond certain loan to values(LTV). (LTV is your mortgage loan devided by the value of your house). Currently the best deals are available for those with a LTV of less than 60%. Roughly the next step is about 75% LTV and then 90% LTV. Above this you will find it very difficult to remortgage.
Regardless of your situation, the easiest way to find the best deal for your current circumstances is to speak to a good independent
mortgage broker. They will explain to you what your options are and inform you of the best rates available to you.