In the Netherlands specifically, there are several reasons to pay extra off on your mortgage.
First, house prices have dropped significantly in the last several years. They are rising slowly now, but it's region specific and you can still borrow more than 100% of the price of the house. Under these conditions, if you choose to sell your house and the outstanding mortgage amount is greater than the value of your house, you are left with a gap (restschuld) to finance. I think the rules have changed recently around this, allowing you to finance this gap with a new mortgage, but this is not a good idea. The tax implications of this are likely to be complicated in the long run and your new house may not cover this gap for some time.
Second, the less you owe on your house, the lower mortgage rates you can get. Mortgages in the Netherlands usually fall into categories based on percentage of the auction price at a foreclosure sale (executiewaarde). If you pay more of your mortgage off, you may qualify for a lower interest rate, possibly making refinancing interesting. This is especially important if interest rates continue to drop but the value of your house does not increase or even decreases.
Third, if you choose to keep your house and rent it out, the banks in the Netherlands have very strict rules on this if you want to do it above board. I've read that some banks require the mortgage amount (NB not the value you may have built up in a linked savings or insurance account) to be less than 50% of the foreclosure auction price (executiewaarde). Also, related to point 2, if you have something other than a linear or annuity mortgage, you will need to refinance to do this as the tax advantages around savings mortgages ([bank]spaarhypotheken) do not apply if it is not used as your own residence.
Finally, if you choose to sell and you are in the happy position of having the value of your house be greater than the value of your mortgage (you have an overwaarde), there may still be some obstacles. Any value you have accumulated in a linked savings or life insurance account is not available until after you sell your house. Extra value derived purely from the difference between mortgage value and sale price may be easier to deal with.
EDIT: As a final note, I've made extra payments on both a "Spaarhypotheek" (linked life insurance) and a "Bankspaarhypotheek" (linked savings account). In one, the principal paid each month reduced and the mortgage lifetime stayed the same. In the other, the principal paid each month stayed the same and the lifetime reduced. In both cases, interest payments were less each month. I would contact your mortgage provider to understand what the expected impact of extra payments will be.