This isn't really a black and white question. I think the amount of money that the UK should pay into the EU after it leaves is totally dependant on what kind of deal they get. If we allow free movement and customs then they should continue to pay a substantial amount. However, if they take the full Brexit I see no reason why they should continue to pay.
Britain should not still pay into the European Union after they exit it because they are no longer a part of the union. If they are solely responsible for their well-being after the exit than they are not responsible for support of the Union. They should have the option to pay in to the union for protection.
Once Britain exits the European Union they should have no further need or obligation to pay into it. Their exit means that they will not enjoy the benefits of being in the EU, and they should also not incur costs for being in the EU. Their exit should cut the financial ties.
If Britain continued to pay into the European Union after the country leaves the institution then it would inspire real anger in the United Kingdom, which could spill very easily into civil unrest. The mood in the country during and immediately after the EU referendum was fairly ugly and even led to acts of brutal violence, including of Jo Cox. The terms of Brexit as understood by the general public need to be met in order to begin reuniting the country. This means that the UK should not continue to pay in to the EU.