tl;dr: For the relevant part of Georgia, Yes, it is run by Democrats, but this is not true for the state as a whole.
The majority of the work in elections in the United States is run by local election commissions, not by the state government (as Trump's quote seems to misleadingly suggest.) These are typically operated at the county level and appointed by county-level politicians. Thus, they are appointed by politicians who tend to reflect the political views of the individual counties, not of the state as a whole.
(For those not familiar with U.S. political systems, 'counties' are the political divisions beneath states, with each state typically being divided into many dozens of counties. According to the U.S. Geologic Survey, the 50 U.S. states plus District of Columbia (the capital district) are divided into 3,141 counties or equivalents of counties. Very large cities may span a few counties, though most counties contain several smaller towns and cities.)
The bi-partisan National Conference of State Legislatures describes the breakdown of work and oversight in running elections in this way (emphasis mine):
The U.S. is characterized by a highly decentralized election administration system. The entities that do the rubber-meets-the-road functions of running an election are typically on the county or city/town level. The state is responsible for certain aspects of elections as well, and the federal government has a role, too. The result is that no state administers elections in exactly the same way as another state, and there is quite a bit of variation in election administration even within states. Each state’s election administration structure and procedures grew organically, as times changed and administering an election became an increasingly complex task.
In general, there are some federal laws governing elections that all states and campaigns must follow. State laws define much more of the nuts and bolts of how elections should be run and a state-level agency typically oversees elections for the entire state and approves the final vote counts. However, the majority of the voting processes are defined by local-level election commissions and nearly all of the work of providing the means to vote and counting of the votes takes place under their direct oversight and according to their rules. They then report results up to the state-level election agency.
So, in a large, urban areas like Atlanta - where every Mayor since 1879 has been a Democrat - yes, most of the election apparatus is, in fact, run by Democrats. Being much more populous than most Georgia counties, these are the counties where the vast majority of vote-counting over the last few days has still been taking place. The less populous counties, which tend to support and be run by Republicans, can typically complete their vote counting much sooner and most of that was done by election night. This is why Trump led Georgia by several percentage points late Tuesday night and his lead has been slowly declining since that time, ultimately switching to a very slight lead for Biden as of this morning (Friday, Nov 6.)
Note that this pattern is true in essentially all states that have any large cities, not just Georgia. The rural areas tend to be very conservative with local governments run by Republicans. Suburbs tend to be center-right with a mix of both Democratic and Republican-run governments, though somewhat favoring Republican-run ones. Large urban areas tend to be heavily Democratic with local governments run almost exclusively by Democrats. In all of those cases, the majority of the election apparatus is run by those local governments.
Edit: Another answer has added some incorrect information, so, to clear up the misconceptions mentioned in the other answer:
The "non-partisan" part of the Chairperson position on election boards in Georgia means that the position is not reserved to be nominated by one party or the other. It does not mean that the people who did nominate the Chairperson aren't from one party or the other or than the nominated Chairperson isn't from one party or the other.
For example, in Fulton County, Georgia's most populous county and the location of most of Atlanta, the election board has 5 members. Two are nominated by each party and the other (the Chairperson) is appointed by the Fulton County Board of Commissioners. Unsurprisingly, the Fulton County Board of Commissioners has a Democratic majority.
In particular, according to a contemporaneous article from the Atlanta Journal-Constitution, current Chairperson Mary Carole Cooney was nominated in 2013 by Democratic Fulton County Commissioner Robb Pitts:
County Commissioner Robb Pitts, an Atlanta Democrat and former city councilman, nominated Cooney to chair the elections board. She spent 22 years as an attorney with the City of Atlanta. For the last 14 years she has been in private practice and since 2005 has served as secretary of the League of Women Voters of Atlanta-Fulton County.
The website of the Registration and Elections Board confirms that she remains the Chairperson. While Ms. Cooney's position does not require her to officially state membership of a party, her past campaign contributions show that she has contributed repeatedly and exclusively to Democrats, including John Kerry, Barack Obama, John Lewis, Jim Martin (who unsuccessfully ran for U.S. Senate in Georgia,) and others. It appears that she stopped contributing to political campaigns after her appointment to the election board. So, based on her own campaign contributions, the party of the commissioner who appointed her, and the fact that she was previously the Deputy City Attorney for Atlanta (which, as previously mentioned, has been run by Democrats continuously since the 1800s,) it seems fair to conclude that she's a Democrat.
This is not at all to suggest any wrongdoing on her part, nor that this situation is unusual. It almost certainly isn't. It's also not to suggest that the elections in Fulton County (or anywhere else) have been unfair. At least on a meaningful scale, that seems unlikely. But that isn't the subject of this question. This question is asking if the election apparatus is run by Democrats. Completely unsurprisingly, in Atlanta, it is. Just as it's almost certainly run by Republicans in the majority of Georgia's rural counties. Elections in the U.S. are run by the local governments and, thus, the people running them normally reflect the political views of the local population.
