Does enabling the alternate RXD function on an AVR GPIO pin disable the pull-up functionality?
The answer is in fact documented in the datasheet in the section called "Alternate Port Functions". 

As you haven't given a part, the example below is for the ATMega162, but according to the datasheet:


  "The overriding signals may not be present in all port pins, but the
  figure serves as a generic description applicable to all port pins in
  the AVR microcontroller family."


...so this information should apply to any microcontroller in the AVR family. 

Referring to the datasheet we find the Alternate Port Functions section starts on Page 68.

At the start of the section we have a general information which explains how the alternate functions work - specifically how the IO ports are controlled by "override signals" which as can be inferred from the name are signals which override various parts of the functionality of the IO port.

According to Table 28 on page 69, for the pull-up resistor, the override signals are PUOE (pull-up override enable) and PUOV (pull-up override value). When PUOE = 1 (override enabled), the pull-up resistor is enabled only if PUOV = 1. When PUOE = 0 (override disabled), the pull-up resistor has normal functionality.



So the task is then to work out what values the PUOE and PUOV signals take when using the RXD alternate function. The answer is given in the various tables following. For the ATMega162, RXD0 is on Port D, bit 0, as per table 38 on page 78.



Now we know which alternate function we are after, lets look for the table referring to override signals for PD0. This can be found in Table 40 on page 80.



Here we can see that for PD0, the values of PUOE and PUOV are RXEN0 and PORTD0 . PUDn respectively. Breaking it down, this means that whenever the UART Receiver is enabled (RXEN0 = 1 - see USART register descriptions), then the pull-up resistor override is enabled. That means the pull-up resistor is now controlled by the PUOV value. We can see then from the equation that PUOV = 1 when (PORTD0 . PUDn) = 1 - meaning when the logical AND of PORTD0 and PUDn is true. This is the case when PORTD0 = 1 and when PUDn = 1.

TL;DR; When the RX alternate function is enabled, we can see from the datasheet that the pull-up resistor is still controlled by the PORTD0 register bit just like in normal (GPIO) operation.