http://ask.fm/dermetfan/answer/107523209834

private Array<AnimatedTiledMapTile> animatedTiles = new Array<AnimatedTiledMapTile>();
    private TiledMapTileLayer backgroundLayer;

    private void animateTile(float delayBetweenAnimation, String tileSet, String tileToAnimate) {
        Array<StaticTiledMapTile> frameTiles = new Array<StaticTiledMapTile>();
        Iterator<TiledMapTile> tiles = map.getTileSets().getTileSet(tileSet).iterator();

        while(tiles.hasNext()) {
            TiledMapTile tile = tiles.next();
            if(tile.getProperties().containsKey("animation") && tile.getProperties().get("animation", String.class).equals(tileToAnimate))
                frameTiles.add((StaticTiledMapTile) tile);
        }

        animatedTiles.add(new AnimatedTiledMapTile(delayBetweenAnimation, frameTiles));
        for(TiledMapTile tile : frameTiles)
            animatedTiles.peek().getProperties().putAll(tile.getProperties());

        for(int x = 0; x < backgroundLayer.getWidth(); x++)
            for(int y = 0; y < backgroundLayer.getHeight(); y++) {
                Cell cell = backgroundLayer.getCell(x, y);
                if(cell.getTile().getProperties().containsKey("animation") && cell.getTile().getProperties().get("animation", String.class).equals(tileToAnimate))
                    cell.setTile(animatedTiles.peek());
            }
    }