Some advice here, before I dissect your map in my giant answer: it would be really, really helpful to anybody trying to evaluate the map if you had at least the latitude lines on it (longitude would be nice, but isn't really relevant to climate). It's also useful if your map is in a known projection when presented for evaluation and/or while editing, even if you intend on a different final version; I would recommend an Equirectangular projection, because those are extremely simple and easily parsed by a wide variety of programs. I'll give you what pointers I can about climate problems (and why your map has other problems), but in the future try to work based on a known map projection for ease of use and evaluation.
First, the problems with the map itself.
Right now, with the tilted map you're using (the latitude lines would probably be diagonal if drawn on it, as I assume the icy areas are near one of the poles), I can't identify several key properties about your world, so I have a hard time estimating how accurate your biomes are. Predicting the ocean currents or wind patterns requires latitude lines, as well as knowing which way the planet is spinning (clockwise or counterclockwise; this alters their direction), and those are both needed to accurately estimate rainfall or temperature.
I can't even be sure where your equator is right now, but based on your scale (another problem: any fixed scale is inaccurate when applied to a map projection, especially on a planetary scope, so that scale will actually change depending on which part of the map you apply it to; just try moving around the world in Google Maps and watch how the scale bar changes even without adjusting the zoom) and assuming that this planet is approximately Earth-sized (meaning a circumference of roughly 40,000km, so about 10,000km along the surface from either pole to the equator), the equator would be somewhere to the southeast of what you have displayed. In other words, you're only showing one hemisphere, and not even all of it, because even taking your map at a diagonal is probably not 10,000km based on your (flawed) scale. If your planet is materially smaller than Earth, it's far enough from Earth-like that you're going to need to estimate gravity and air pressure as well, making your job even harder.
Now, to the climate.
First, your jungle biomes are hideously out of place. Tropical conditions are essentially defined as high monthly temperatures (18C or greater) year-round with at least moderate rainfall (without enough rain, you get deserts). The tropic lines (Cancer and Capricorn) are the outer limits of where you'll find tropical conditions, as these only really appear near the equator; beyond equatorial regions, seasonal variation becomes too great and you end up with temperate or continental climates.
As such, having jungles anywhere even remotely close to taiga/snow forest (which you need to clarify; snowy forests are mostly taiga, and tundra doesn't have trees if that was what you meant) is utter absurdity; taiga requires temperatures below 0C several months of the year, and often the warmest month will not reach the 18C average that is considered the minimum for tropical conditions. Assuming an Earthlike axial tilt (about 23 degrees; this is where you find the tropic lines), there's going to be at least 20-25 degrees of latitude between tropical regions and taiga regions, probably more. I'm referring to the Koppen climate classification, if you want to read up in more detail (tropical biomes make up the A group, with tropical rainforests as Af; taiga would be the colder continental areas, mostly Dfb or Dfc).
Your rivers are also flawed. As sphennings has already said, rivers flow downhill, not up, so any river heading into a mountain range is wrong; they might start in a mountain range, but barring some very deep valleys that you don't have marked, they won't flow through them. They also converge into one another except at deltas; rivers have on occasion been known to split, but it's quite rare and not a lasting phenomenon (one side of the split will draw more water and erode faster, and eventually the other side will simply dry up as the full flow goes to the other branch), so don't make them spread out as they go until right before they hit the ocean.
Rivers flowing out of lakes follow a similar principle, which you should keep in mind. Multiple rivers might flow into a single lake, but only one river should be flowing out from it. That southern continent (all forest/grassland) has a particular example; I count three rivers connected to it, and all three end in the ocean, which is unrealistic in the extreme. Two of those (I would suggest the southern two, because the tributary river that merges into the northern one has a direction that suggests it's going to the ocean) should be flowing from springs or mountains into the lake, and only the third should be continuing on to the ocean.
The southeastern desert island is also horribly, horribly wrong. You have a lake surrounded by mountains, which are surrounded by desert; that lake will not last, and you certainly won't have nice grasslands around it. It's going to look more like the interior of Australia than anything else. That interior mountain range is also suspect, from a tectonics perspective; isolated volcanic mountains can pop up almost anywhere due to hot spots, but ranges are typically formed by plate collisions, which means most ranges are near coastlines. There are some other cases where they result from ancient collisions (North America's Rocky Mountains; the west coast has two separate mountain ranges) or the collisions are inland (see the Himalayas), but your inland mountains don't really fit either bill.
Precipitation there comes from moisture blown in over the ocean, but as the clouds rise up the mountains they will rain themselves out on the opposite side from the lake; this is known as orographic lift or rain shadow. When the clouds get over the mountains to your lake, they will have little moisture left, and since it gets warmer as the clouds drop (preventing condensation and rainfall), that moisture will not fall as rain near that lake. You'll most likely get rivers flowing out from the mountains to the northeast, towards the non-mountainous coastline. The southwestern side (as seen from your map) of that continent would be bone-dry apart from the immediate coastline, due to the outer mountain range that would suck up any rainfall.
Without latitude lines, I can't predict what ocean currents would flow around that continent, so I can't tell you for certain if the coast would ever receive any real rainfall; if you end up with cold currents flowing past the coast, it's going to turn out extremely dry (you wouldn't even get the rain shadow effect, most likely, so you'd get lots of fog and clouds but next to no actual precipitation: see northwestern Africa). My best guess (based on my guess that it would probably be a warm current) is that the northeastern side will probably end up a lot like either Egypt or India, with distinct wet and dry seasons; either floodplains with one or more rivers that flood annually like the Nile, or a more general monsoon that brings lots of rain in some months and is dry for the rest. The latter scenario would probably produce savannah (Aw/As climates in the Koppen classification) and/or steppe (BS climates) biomes, the former would be a desert (BW climates).