I have a hard time answering this question because I'm stuck on how to make ice not float. Are we looking to rework the laws of physics themselves, or a work around within physics to justify it?
The easiest way to make this happen would be to assume a world where water was not the most common liquid. However, the question of "how would the world develop if some other liquid was the building block of life instead of water" would be far more important then this one.
Thus lets assume that were talking about water. The easiest way to keep ice from floating would be to decrease the density of water by contaminating it.
Going with this approach you would also have significantly lowered the freezing point of water. Thus there would be far less ice being formed to begin with. When the ice did begin to be produced it would sink down, allowing more ice to form. Presumably the temperature of water below the surface would be lower then above the surface, so the ice would continue to sink. As it sunk pressure on the ice would increase, which would further encourage it to freeze. The ice would not unfreeze easily at this point!
Thus, despite the freezing point of ice being much lower in these oceans, I imagine you would have huge frozen patches of ice in the ocean. Since once the ice sank the pressure and increasingly lower temperature would conspire to keep it sunk. I imagine that in fact most of the ocean would be frozen this way, only the very top of the ocean, where sunlight penetrates enough to keep temperature's above freezing, would be liquid. Luckily since the actual freezing point of water would presumably be much lower the ocean would never freeze entirely. Even during winter It would be too warm near the poles for water to freeze.
To add to this ice would be formed under the surface instead of at the surface of water. As of now in a perfectly clean lake that is frozen over will have a temperature of 4 degrees Celsius at the bottom. This is because 4°C is the temperature at which water is the most dense, after this point good old hydrogen bonds kick in and make colder water start to float back to the surface( quick tip, if your in chemistry class and asked why water does something funny assume it's hydrogen bonds, your ace every test). Thus the temperature of even huge bodies of water are pretty well regulated, no matter how deep you go your not getting colder then 4°C. Without this principle the water below the surface will be much MUCH colder since there is no source of heat and nothing to encourage the warm water above to sink lower.
This also means that water will 'churn' less, warm water won't naturally be pulled down further into the ocean. This probably means less oxygen in the depths, I imagine part of the oxygen that low is transferred from warm water on the surface sinking. I don't know how significant this effect is, probably minor compared to the effect plankton has. It's a moot point though, since water won't get nearly as deep before ice forms.
You would also have Ice poles, locations where the temperature of water is regularly lower then freezing where you have a solid 'pole' of ice from surface all the way to the ground. These poles would expand out radially until they reach a point where the average temperature is warm enough that surface water doesn't freeze. Then ice will slowly drop as you expand further out, from being on the surface to being 1 inch below to 3 inches below surface to 10, etc. etc. A slow steady drop as you expand radially out from the ice poles.
At warmer areas your have most of your ocean frozen. Those parts close enough to sunlight, and thus heat, will stay liquid water. I'm not sure how far you will have to go below the surface before you start hitting ice. I'm too lazy to do the math. However, I don't think it will be too deep. Lakes will probably be ice-free during the summer, but once you get deeper then your average lakes that ice will form due to lack of heat from sunlight. Your ocean life will be far less common.
Small lakes will freeze entirely during the winter, but perhaps not large ones. The water itself works like a huge heat sink, storing large amounts of heat. It takes time for that much heat to be expelled. And it takes a significant amount of heat to freeze water into ice to begin with. Thus large lakes may only 'mostly freeze' because the temperature warms up before the lake has had time to freeze over properly. To give an example of this concept, in reverse, imagine a pile of snow created by a snow plow. After it warms up above freezing snow will start to melt, but the snow in the snow pile could take a week or more to melt in full despite it being above freezing because the snow insulates itself. The same principle, in reverse, means it would take quite awhile for lakes to freeze. When the do freeze they would freeze from the bottom up, so large lakes would still stay liquid for most or all of winter.
However, any pond that is small enough to freeze over entirely will kill most animal life, though I stress most. Currently fish and other lake animals live underneath the ice where it's still liquid, the liquid ice on the top prevents the rest from freezing by insulating it so well. With the entire lake frozen it would seem that all creatures would die, but frankly that doesn't give evolution enough credit. Instead we would have fish that lay eggs that can survive freezing and which hatch after the lake thaws, fish that can live in shallow 'mostly frozen' lakes by hibernating, all kinds of ingenious evolutionary tactics to survive. Still, I would expect far less diversity in such ponds.
If you want a better analysis on the effects of this I would really encourage you to ask here: https://what-if.xkcd.com/ This is exactly the sort of question he answers, and he will do a far better job then I would :).