The Atlantic has an interesting summary of the recent literature focusing on the negative effects of emotional intelligence. Essentially, emotional intelligence not only allows for better interpersonal relations and cooperation but also a greater ability to manipulate others.
One of the most interesting examples that the article gives is that, in non-emotional work (data analysis or car repair rather than counselling or teaching), there is actually a negative correlation between emotional intelligence and job performance (see here for the review article). The Atlantic article proposes that emotional intelligence distracts people from their work in these types of jobs: people spend their time reading their colleagues rather than their spreadsheets.
I have an alternate explanation that should probably be considered. While emotional intelligence may not make you better at low emotion jobs, it probably makes you more likely to be promoted or hired (conditional on prior job performance). If this is the case, then the negative correlation is simply the result of selection into jobs on the basis of emotional intelligence (due to bosses liking the employee or good interview performance).
Essentially a person with low emotional intelligence needs to be better at their job than a person with high emotional intelligence to get hired for the same position. It certainly fits better with my anecdotal observations than people being distracted by their emotions (surely people with more emotional intelligence need to expend less energy on reading those around them).
This hypothesis is also compatible with the finding that emotional intelligence is associated with better job performance in emotional work. In emotional work, emotional intelligence is a good signal for job performance (indeed it may be better than formal indicators), so promoting someone based on it probably improves the job/employee fit.
I've not read the literature in much depth so I'd be interested to hear if this hypothesis has been tested somewhere.