Does Trump's federal hiring freeze affect academic jobs in any way?
It's complicated.

In the United States the interplay between federal employment and academic positions isn't very clear-cut. This has resulted in unusual situations where members of the same department may be employees of a number of different entities and may be either federal employees, university employees, or organization employees (ex. Harvard-Smithsonian Center for Astrophysics (CaF)).

The text of the memorandum includes the following:


  As part of this freeze, no vacant positions existing at noon on
  January 22, 2017, may be filled and no new positions may be created,
  except in limited circumstances.


and


  Moreover, it does not limit the hiring of personnel where such a limit
  would conflict with applicable law.


The "except in limited circumstances" is the relevant bit because it means it means that open (or opening) positions may still be filled. From a pragmatical standpoint, it means that if a job positing went up after January 22, 2017, odds are someone can be hired into it. 

Another line that is relevant is the following:


  Contracting outside the Government to circumvent the intent of this
  memorandum shall not be permitted.


This means that organizations that hire contractors are likely to put new contract roles on hold unless the position is vital to national defense. This may impact national labs, but is heavily dependent upon what the line of funding for the contract is. Be sure to ask about this if you are interviewing for a role.

Since this memorandum does not impact grant funding, academic and research positions that are paid through an existing line of funding are exempt. 

In terms of actually conducting a job search. If you are on USAJOBS you should pay close attention to the details of the posting since some since some organizations like the CaF advertise for non-federal employees on there. These jobs would not be impacted by a federal hiring freeze. Likewise, positions that are in exempt from hiring freezes due to legislation (ex. medical professionals) are also not impacted. Positions at state and private universities would also not be impacted, and would unlikely to be advertised on USAJOBS.

Generally the most basic rule of thumb is that if you are applying for a job that is listed under the General Schedule there is a high likelihood that the hiring freeze applies.