Short answer: Neither the IRS nor the Census Bureau employ subject-matter experts of any kind to decide what is or is not a church or religion. The Census Bureau just doesn't collect that data. They do re-publish data from an academic survey, but that survey just aggregates responses to open-ended questions (avoiding the problem entirely). The IRS must make a determination what is a church, but for their purposes specific theological content is irrelevant.
The Census Bureau Does Not Collect Religious Affiliation Data
The Census Bureau stopped collecting data on religious affiliation in 1957. According to the Bureau they originally collected this data as part of a survey independent of the decennial census, called the Census of Religious Bodies.
You can see that there is no religious affiliation information in the demographic data by using their interactive population tool.
Finally, federal law prohibits the Census Bureau from asking questions about religious affiliation. This law was passed in 1976.
The Statistical Abstract
The Census Bureau does publish the Statistical Abstract, which includes religious and denominational affiliation data. If you download one of their Excel files, it includes a note explaining what they do.
The data comes from the American Religious Identification Survey, which is performed by Trinity College's Institute for the Study of Secularism in Society and Culture. Both of the PIs are demographers with a background in public policy.
The survey uses open-ended questions when asking about religious and denominational affiliation. Respondents are not prompted with a list of acceptable answers to choose from, they are just asked about their affiliation and then the results are aggregated.
The notes are contained in Excel files. To save anyone the trouble of downloading them and finding the notes, I have copied them below.
The American Religious Identification Survey (ARIS) 2008 replicated the methodology of ARIS 2001 and NSRI 1990. The three surveys are based on a random-digit-dialing telephone surveys: 54,461 interviews in 2008, 50,281 interviews in 2001 and 113,723 in 1990, all in American residential households in the continental U.S.A (48 states). Respondents were asked to describe themselves in terms of religion with an open-ended question. Interviewers did not prompt or offer a suggested list of potential answers. Moreover, the self-description of respondents was not based on whether established religious bodies, institutions, churches, mosques or synagogues considered them to be members. Quite the contrary, the surveys sought to determine whether the respondents themselves regarded themselves as adherents of a religious community. Subjective rather than objective standards of religious identification were tapped by the surveys]
Note about non-denominational responses:
Because of the subjective nature of replies to open-ended questions, these categories are the most unstable as they do not refer to clearly identifiable denominations as much as underlying feelings about religion. Thus they may be the most subject to fluctuation over time.
And about agnostics:
Atheist included in Agnostic.
The IRS is in charge of enforcing tax policy by appropriately collecting taxes. The Internal Revenue Code uses the word "church" without defining it. This creates a problem for the IRS, as they have to enforce a rule that isn't adequately defined. To resolve this problem they have a list of criteria that help them decide what is a church and what isn't. The list is here, but importantly there is no consideration of what their beliefs are.
Short answer - No they do not employ theologians. The IRS does not take into consideration the content of a religion's beliefs when deciding what is a church.