(Downvoters please note that the question is about people considered celebrities in the UK. If what you're interested in is people considered celebrities in the USA, please skip to Oddthinking's answer. There's nothing unusual about the number of American or world celebrity deaths in 2016)
There's clearly no "curse". There are objectively measurable indicators that can be compared between years, and these do appear to suggest a sharp increase in 2016, approximately double that of recent years, for deaths of people considered celebrities in the UK, without any one obvious apparent causal event (such as a helicopter crash). That said, these are still small numbers (lower end of two figures), and occasional spikes, anomalies or clusters in small data sets aren't unusual and don't normally require a causal explanation.
The same evidence shows the number of deaths of people considered celebrities was rising already, which can be easily explained by mass media reaching a certain age.
There's an objectively measurable of "celebrity" deaths - obituaries written in news publications. Note that these are typically written while a celebrity is still alive then kept on file, so an "obituary use" represents an editorial team's judgement that an individual is a celebrity of note made well before and independent to any public reaction to their death, and most likely written well before 2016 (therefore not a self-fulfilling response to the so-called "2016 effect"). It's obviously not a perfect classification (no human judgement based classification is), and its quality depends on the quality of their editorial process, so I've taken three examples from three different respected UK publications with different approaches:
In an article on precisely this phenomenon the BBC quote data about the numbers of obituaries they've written in January-March for the last 5 years. This suggests an upward trend and a 2016 spike: 24 in 2016, 12 in 2015, 11 in 2014, 8 in 2013, 5 in 2012.
[Note - these figures were added to the question after I included them in this answer]
They discuss whether this could be explained by expansion of the facility of the BBC to write and publish obituaries, and find that the number of celebrities with obituaries written and kept on file in case of their death is increasing, but at a rate of "a few more every week", following a process that hasn't significantly changed in 10 years, with no one event that would explain a doubling from 2015 to 2016 other than random clustering.
This therefore addresses the question:
Are the celebrities really dying faster than usual, or are we just reporting on them more?
The BBC have had a consistent policy for 10 years which is based on steady inflation through a team actively increasing the number of obituaries on file every week, but 2016 is an anomaly that bucks the trend.
(The Telegraph is a broadsheet sometimes described as the UK's "newspaper of record)
The above article reports:
the Daily Telegraph maintains a gallery of famous people who have died, and updates it throughout the year. Up to this time in 2014, the number of those in the gallery was 38. By this time last year, the number of people in the gallery was 30. This year, the number is already 75
Again, the number is roughly double that of recent previous years.
(The Guardian is a broadsheet and a partner of the International Consortium of Investigative Journalists)
The Guardian offered some measurable figures in terms of the number of deaths which they ran as the main headline of their print edition: 7 so far (one of which, being a sportsman, was printed in the sports section but as the main back-page headline).
I can't find a direct comparison to equivalent periods in other years, am still looking, but it's discussed as being remarkable:
The tally of deaths is significant, and 2016 is not yet four months old.
They offer one more observation:
The Guardian’s obituaries page, according to its editor, is no longer able to fit in all the public figures who would have historically figured on the page because of the rise in high-profile deaths
In the Guardian's case, like the BBC, this therefore suggests that the number of obituaries they want to publish in 2016 is driven by events, not an editorial decision: events are causing them to exceed their physical limit.
With this, they offer a partial explanation for the upward trend:
The nature of fame and celebrity has changed radically in the past decade...
As well as actors, musicians, royalty, sporting figures, politicians and the like, there are now hundreds of reality TV stars, comedians, game show hosts, mavericks and people famous simply for being famous. That means there are far more celebrities whom more people will have heard of when their number is up.
Deaths among the general (non-celebrity) UK population
Or - chillingly - is everyone dying off faster, and we're just hearing about the famous ones?
Since there's no celebrity death spike outside the UK, we only need to look at UK mortality statistics here. The UK Office for National Statistics publish weekly death statistics, and offer a comparison to the last 5 years week-by-week. They haven't yet published an analysis comparing 2016 to other years, or age-standardised rates, but a simple look at the raw numbers of deaths shows that the increase between 2016 and the previous 5 years appears smaller than the increase between 2015 and the previous 5 years (around 3% compared to around 7% from 2015).
There's therefore no reason to think there's any particular UK trend at work here, and no reason to think that the so-called "curse" widely discussed in the British media is anything more than the kind of random spike that can happen with any small variable data set.