- I don't have any special plans for this weekend.
Q1: What is the meaning of 'for' here? Q2: What is the part of speech of 'for this weekend'? Does it modify 'any special plans'? Or does it modify 'don't have'?
In OP's context, including the preposition strongly implies that for this weekend should be interpreted as an "adjectival" element modifying the plural noun [special] plans.
Without the preposition, this weekend is an "adverbial" element modifying the verb [to] have [plans].
In practice, it doesn't really make any difference to the overall meaning in the exact cited context, but there could be a difference in, say,...
1: I paid the bill for last week
2: I paid the bill last week
...where exactly the same syntactic distinction applies.
In #1 it's the bill that relates to [goods / services delivered] last week - which might be paid for later.
In #2, it's the act of paying that happened last week, where last week adverbially modifies the verb-based action [to] pay [the bill].
It may help to note that it's not possible to say I am paying the bill last week, but it is perfectly natural to say I am paying last week's bill or I am paying the bill for last week.