Edit in JSFiddle

var map = $('#map'),
    m = {};    

map.tinyMap({
    'center': {'x': '25.04151536540612', 'y': '121.56354904174805'},
    'zoom': 14,
    'marker': [
        {
            'addr': ['25.04151536540612', '121.56354904174805'],
            'text': '25.04151536540612, 121.56354904174805',
            'event': {
                'mouseover': function () {
                    var mm = map.data('tinyMap');                    
                    this.infoWindow.open(mm.map, this);                    
                },
                'mouseout': function () {                    
                    this.infoWindow.close();
                }
            }
        }            
    ],
    'event': {
        'idle': function () {
            
            // 取得 tinyMap instance
            m = map.data('tinyMap');
            
            // 取得 markers 及其 infoWindow
            var marker = m._markers[0] || {},
                infoWindow = marker.infoWindow || {};            
            
            if ('function' === typeof infoWindow.open) {
                // 開啟 infoWindow
                infoWindow.open(m.map, marker);                                
            }            
        }
    }
});

// Switch button
$('#switch').on('click', function () {
    
    // 取得 tinyMap instance
    m = map.data('tinyMap');
    
    // 取得 markers 及其 infoWindow
    var marker = m._markers[0] || {},
        infoWindow = marker.infoWindow || {};
    
    // 判斷 infoWindow 是否已開啟
    if (!infoWindow.getMap()) {
        infoWindow.open(m.map, marker); // 開
    } else {
        infoWindow.close(); // 關
    }
});

<button type="button" id="switch">Switch</button>
<div id="map"></div>    

#map{width:100%;height:300px}