This question is rather weird. The questions should be why is it difficult to collect the results quickly and accurately. Because they were collected quickly e.g. in 2012, but took 16 days to certify, eventually producing a different winner than that announced on caucus day. The reasons, back then, were mainly typos.
There is no single reason why things went wrong in Iowa, but here's a historical summary of what did go wrong:
1976:Iowa Democratic officials ignored a rule in calculating delegates, resulting in exaggerated projections of delegates for front-runner Jimmy Carter. The Des Moines Register discovered the problem days later.
1980: The problems with the 1980 counts are legendary.
It appeared on caucus night that George H.W. Bush scored an upset victory, beating Ronald Reagan by as much as 6 percentage points.
But computer problems kept 165 mostly rural precincts in which Reagan figured to do especially well from being included in the tally. Two days later, the party’s re-examination showed a 2 percentage-point margin for Bush. CBS News had it even closer, with Reagan leading by less than 1 percent.
The final numbers represented 94.4 percent of the precincts — 142 precincts never reported their results or didn’t hold caucuses, according to Drake University political science professor Dennis Goldford, co-author of “The Iowa Precinct Caucuses: The Making of a Media Event.”
1984: The News Election Service, funded by a consortium of national TV networks and the Associated Press, was created to gather raw vote totals rather than the delegate equivalents that party officials had provided since 1972. But Democratic Party officials refused to cooperate, and the news service managed to tally only 74 percent of precincts.
The party’s own counts proved unreliable: Party leaders were sure Walter Mondale had come in first, but they weren’t certain where other candidates had finished. Many votes were never turned in.
1988: Democrats were still at odds with the news media. Questions were again raised about the validity of the News Election Service results, which were based on just 70 percent of Democratic precincts, Goldford wrote.
Republicans, however, continued to work with the News Election Service to ensure accuracy and legitimacy, and Bob Dole’s nearly 13 percentage-point lead was based on 98 percent of the GOP precincts counted.
1992: Again, Democrats refused to share caucus vote totals, only delegates won by each candidate. The GOP didn’t hold a vote to save its sitting president, George H.W. Bush, any embarrassment from challenger Pat Buchanan.
1996: Republicans decided to do no official party count. Instead, the tabulating was handled by the Voter News Service, which replaced the News Election Service. Still, some GOP candidates thought the VNS results showing Dole first and Buchanan second were flawed.
2008: Republicans handled their own count and experienced some data entry problems. Mike Huckabee won, with Mitt Romney in second. Democrats reported a win for Barack Obama, with John Edwards and Hillary Clinton nearly tied for second.
2012: Iowa Republican officials moved their tabulating center from party headquarters to an undisclosed location to ward against hackers and protesters. County officials reported the precinct votes by phone with live call-takers or logged in to a security-code-protected website.
So yeah, counting is easy, except when people are involved. If I were to summarize from the above, it seems that party-ran caucuses don't have a great track record, but that may be a biased conclusion without comparing it with elections that were held at the same time. It seems somewhat less controversial that finding the right level of outsourcing for party elections has be a tâtonnement process, in Iowa at least.
The NYT now has a more detailed article on this year's problems. The updated instructions don't appear to have been followed everywhere, there were typos involved again, and even data copied in the wrong columns. Because more data was reported (electronically) than in previous years' caucuses, the problem was somewhat different:
Just about every election night includes reporting errors. They can be difficult to identify, but can often be corrected during a recount or a postelection canvass. This year’s Iowa caucuses are the reverse: Errors are now easy to identify, and hard to correct.