There are innumerable demonstrations all the time at the centre of Athens, against... everything (many strikes through the year too). Even a few persons who protest are able to make the traffic tangle more chaotic (if it isn't already the worst it can be...)
Sometimes, violent actions can take place during the protest marches (e.x. don't be at the centre of Athens on 17 November...)
..and you are near Syndagma (in particular) you can expect to see lots of police.
You may not (hopefully) see anything much else.
Whilst I was visiting the farmers (so I was told) were demonstrating. Riot police in full gear were in evidence for all the days of my visit, mostly standing around and smoking but on occasion closing or blocking Mitropoleos or adjoining roads.
If there are no problems, then you can just wander around as normal.
Take photographs with some caution and don't make it obvious.
If it looks as if trouble is beginning then I'd advise you to remove yourself pretty speedily. Bystanders can very easily get sucked in (exactly the same advice would apply in the UK or anywhere else).
If you are staying at a hotel in or near Syndagma you will hear the music/loudspeaker speeches etc from demonstrations. I didn't find it a problem as they had all gone home by 6pm.
It seems that demonstrations occur quite often in central Athens, and I almost got involved in once when I was walking towards Syntagma Squre. The demonstrators put up their flags and people started to gather around the square
Be aware that it's no fun being in the middle of a demonstration because when the demonstration police arrives, you might get beaten or hit by tear gas. So, I headed out of there ASAP and got safely into the metro instead
Mid June 2011. It's happening only at Syntagma square exactly in front of the Parliament building. The people are not happy with their government over the nation's huge debt. Every moment there'll be people there but it's safe, unless perhaps if it's really crowded - like happened few months ago. Special polices are around too. Other places are not affected. We passed through them and took pictures.
Another thing I’d like to mention is about the strikes that occur in Athens.The city has a tradition of strikes due to the liberate and restless character of the Athenians(but sometimes indifferent for the fuss that a strike causes to the rest of the citizens)and during winter a large number of strikes take place.A very unpleasant and...smelly strike was that of the garbage collectors.For many days the streets of Athens were teeming with…garbage.If you fall on a strike you don’t have to worry as the vast majority is peaceful.I just mentioned it for the traffic jam it can cause. The picture is from the great strike of Spring 2001,against the new pension-law.Don’t worry!Most strikes are far much smaller…
I came across a very useful, accurate and always up to date link to help all members of VT, tourists and travellers to Athens, get updates in case of up coming strikes in the city centre.