I would like to rid myself of a cursed weapon (whose curse includes the traditional "you may not unequip this weapon" aspect), but don't have access to Remove Curse. Can I turn the cursed weapon into my pact weapon, dismiss the item, and then make a new weapon my pact weapon to get rid of the cursed weapon without Remove Curse?
The simple answer to this is that if the rules don't specifically state that bonding/dismissing a pact weapon can break curses then it does not break curses. Besides, if curses could easily be removed by meta gaming like this then they would lose all meaning. Curses are supposed to have impact.
When the PHB describes dismissing a magic item that has been transformed into your pact weapon, it says:
You can dismiss the weapon, shunting it into an extradimensional space, and it appears whenever you create your pact weapon thereafter.
If you bond the weapon as your pact weapon then it is your pact weapon. There is nothing here that could break a curse.
If you bond to a different weapon or you perform a ritual to remove your bond to the cursed weapon then
The weapon appears at your feet if it is in the extradimensional space when the bond breaks.
So not only do the rules say nothing about breaking curses, they specifically state that the weapon always returns to you. Again, nothing here that could break a curse.
Or more accurately: no better than you could through any other means. If a cursed weapon you're using doesn't have any rules that say otherwise, you may be free to call and dismiss it as suggested, but you could also just set it on the ground and walk away from it.
Much more likely, you're dealing with a cursed weapon that has some rules that prevent you from getting rid of it. For example, the Berserker Axe has the following:
Curse. This axe is cursed, and becoming attuned to it extends the curse to you. As long as you remain cursed, you are unwilling to part with the axe, keeping it within reach at all times. You also have disadvantage on attack rolls with weapons other than this one, unless no foe is within 60 feet of you that you can see or hear.
Being "unwilling to part with the axe" and needing to "keep [...] it within reach at all times" would prevent one from dismissing the axe into extradimensional space using your Pact of the Blade feature, although it would encourage you to bond with it as your pact weapon, in order to return it to you immediately, were it ever forcibly taken from you. However, no part of the Pact of the Blade feature explicitly allows the feature to break curses, so no combination of effects that the feature grants can rid you of the cursed weapon. You'll need to find a different way.