Does The Bridge on the Drina provide an overall picture of ethnic/religious tolerance or ethnic tension in which violence lurks under the surface of inter-communal relations?

To a large extent it provides both, and it keeps them in constant tension throughout the story. This is just one of many factors that makes The Bridge On The Drina such an important book.
What we see throughout is the way in which violent conquest in history brings about immense change: change both good and bad, change that more often than not is completely unpredictable. The Ottoman Empire has just conquered the Balkans; the Ottomans have a fearsome reputation which the local villagers know all too well. They carry out a barbarous practice, a blood tribute, whereby they kidnap Christian boys and take them back to Turkey to be raised according to the local customs and traditions.
Yet Mehmed Pasha, as he eventually becomes, does not forget his roots. Understandably, he is still deeply traumatized by what happened to him as a child. But at the same time, he wants to reconnect with his childhood in a positive way. In ordering the construction of a bridge over the river Drina, he is symbolically rebuilding the intimate relationship with both his mother and his motherland. The symbol of the bridge is an important element in the story, as it provides a means of linking different cultures, religions, and ethnicities together, whatever turmoil and conflict may grip the world around it.
But Andrić doesn't shy away from addressing the intermittent conflicts that break out between different groups over the course of centuries. Even the titular bridge, that great, shining symbol of a link between different worlds, faiths, and cultures, is mired in ambiguity itself. The bridge is built by serfs, little more than slave laborers, forced to toil in the most appalling conditions imaginable. And when they stage a strike against their Ottoman overlords, they are brutally suppressed. In the central symbol of the bridge, then, we can see that for the author ethnic tension and tolerance are by no means clear-cut and that they can, to a large extent, co-exist, albeit uneasily.
The construction of the bridge also forces us to ask uncomfortable questions about how ethnic tensions can arise in the first place. The Ottomans see themselves as ethnically and culturally superior, bringing the benefits of civilization to a remote backwater. The bridge is not just a potential symbol of peaceful co-existence; it also stands as a reminder of the imperial power often needed to advance civilization.
The perspective of the Serbs is understandably different. They groan under the lash of Ottoman oppression. They see the bridge as an imposition upon their land by an alien power, another unwanted manifestation of a foreign invasion. They try to sabotage the bridge's construction, using acts of violence to destroy something that was itself an act of violence against Serb culture, tradition and national self-determination.
Over time, the Ottoman Empire weakens in the face of nationalist uprisings and is forced to retreat from the Serb lands. But one form of imperialist domination is merely replaced by another, this time that of the Habsburgs. Crucially, the village of the story remains at the heart of a multi-ethnic empire. On the face of it, people seem to get along—mainly, though, because they have no choice. They simply must go about the business of making a living. Yet underneath the surface, the threat of inter-ethnic violence and conflict is never very far away.
Once again, the bridge itself is skillfully used to symbolize an important element in the story. It's noticeable that the bridge becomes a focal point of acts of violence that reflect in miniature the wider conflicts in this teeming, ever-changing multicultural society. Not just acts of physical violence, either; the bridge becomes a place of violent disputation in which intense debates concerning nationalism and ethnic identity take place.
And yet despite this, the locals, whatever their religion, ethnicity, or nationality, do still look upon the bridge as the still point of a turning world, a place of stability and civic pride amidst the gathering storm of war. It is noteworthy that when the Drina floods, as so often happens, the villagers put aside their differences and come together to help each other in the midst of adversity.
But there is nothing remotely sentimental about this; Andrić is much too good of a writer for that. He knows that each ethnic group has, to some extent, appropriated the bridge as its own; it has become a blank canvas onto which everyone can project their own wants, hopes, dreams, and desires. So long as it exists, it can to some extent serve as a reminder of common humanity and so keep ethnic tensions in check. But once that bridge is no more, then the tenuous thread that held everyone together will also be destroyed and, with it, the chance of any meaningful reconciliation.