This year has been a quiet year for tourism and the upside of that certainly in Istanbul is that accomodation is cheaper, there no hideously long queues at attractions and one can look around more easily.Security at both airports and on the streets is much tighter and more visible than in the past too and the city can be regarded as being as safe if not safer than any other major world city.
Republic Day will have no affect on you as a visitor other than the fact that banks are closed that day.But exchange offices will work as per usual as will ATM's.As 29th is a Saturday banks would be closed that day normally anyway so no difference for you there.All attractions,museums etc.are open to their normal working days and hours as are all public transport forms.
Special events tend to be of an official nature,i.e wreath laying at memorials,people visiting the graves of family members of were soldiers to pray for them,school parades etc.,things like that.
It is highly unlikely there will be demonstrations or riots that is for sure.
Usually there are fireworks over the Bosphorus but I don't know if that will happen this year since the coup attempt. On the one hand people will want to mark Turkish independence more than ever this year but on the other hand out of respect for those whom died during the coup attempt any celebrations could be more muted than usual.Hard to say and nothing has been announced in the press yet regarding this but as soon as I see anything I'll post up the info.Certainly Victory Day on 30th August was much more lowkey but that was very soon after the coup attempt so we need to wait on this one.
Oh and you can be sure that regardless of whatever else may or may not happen the whole country will be awash with Turkish flags :)