Cursed weapons often have a flat bonus such as a +1 associated with them, however the negative effects of a cursed weapon tend to only be associated with being attuned to said item.
Could a player (who has previously attuned to a cursed item, then had the attunement removed by the Remove Curse spell) continue to use that cursed weapon and benefit from the flat +1 bonus?
In true 5e fashion, you have to read the description of the individual item to know how it works. In the way of general rules we do get:
Without becoming attuned to an item that requires attunement, a creature gains only its non magical benefits, unless its description states otherwise. (DMG 138)
but that just highlights the existence of exceptions. Also, the rules on cursed items (also on DMG 138) do not say anything about their effect working differently in this regard, you are either attuned or not.
Let's see an example for both options. The description of the Berserker Axe states:
You gain a +1 bonus to attack and damage rolls made with this magic weapon. In addition, while you are attuned to this weapon, ... (DMG 155)
My interpretation is that this marks it as an exception to the rule, giving you the +1 bonus without attunement. There would be no need to add the part about attunement otherwise. However, there might be some debate about how explicit should the stating of such an exception be, and I recommend reading this answer for some perspective before you make up your mind about it.
The issue is much clearer in other cases, for example of the Dwarven Thrower:
You gain a +3 bonus to attack and damage rolls made with this magic weapon. It has the thrown property with a normal range of 20 feet and a long range of 60 feet. When you hit with a ranged attack using this weapon, ... (DMG 167)
There is no additional mention of attunement in the description, so the default rules for items with attuenment are in effect, making all properties tied to attunement, including the flat bonus.
By default, benefiting from any bonuses of a "requires attunement" item requires attunement, but it depends on the description, as Szega's answer says.
I disagree with Szega's interpretation that "In addition, while you are attuned to this weapon, ..." implies that the Berserker Axe bonuses are an exception that don't require attunement. I would expect an exception to be written like this:
You gain a +1 bonus to attack and damage rolls made with this magic weapon, even if not attuned. While you are attuned to this weapon, ...
The actual wording is compatible with the default assumption of needing to attune to get the benefits. It means:
You gain a +1 bonus to attack and damage rolls made with this magic weapon [while you are attuned to this weapon]. In addition, while you are attuned to this weapon, ...
However, I would say it is badly written, and other people have interpreted it the way Szega did:
Do magic weapons that require attunement count as magic (for Resistances) even if you are not attuned?
