Can oil be replaced with yoghurt in a cake recipe?

Short answer - not really. Fat is an essential component in any cake, and milk just isn't very fatty - about 5% for whole milk. You can make cakes with milk, but they require totally different recipes: you can't simply substitute milk for oil.

Bear in mind that you're distributing the cup of oil throughout a whole cake, so that any one slice will only have a fraction of the oil. I assume you'll also be sharing the cake with others, so you'll be 'spreading the calories' somewhat.

You should be using a neutral oil like canola (rapeseed) oil anyway, as it has a relatively mild flavour. A popular alternative these days is to substitute all but a couple of tablespoons of the oil with apple sauce (really), but this can be hit and miss and doesn't work with all recipes - it usually works best with things like quick breads (scones etc).

Of course, there's also the point that cakes are supposed to be a treat, and they'd be less of a treat without the fat, in which case why bother? If you want to be healthy, make a salad. If you want a treat, make a cake and don't worry about the contents too much.

It's a popular suggestion to replace oil (yes, a whole cup of it is common in cakes and quickbreads) with applesauce. I've done so and liked the results. I've also used nonfat but sweetened yogurt (a caramel flavour to be specific) with great results in a cake made with zucchini and raisins, and now do that every time I make that cake.

I think you'll run into trouble with milk. Part of the purpose of the oil is to keep the flour from finding other flour and getting breadlike. So you have to experiment a little, and the successful experiments always seem to be something sweet - not all the way to honey or corn syrup, but at least slightly sweet.

A great substitute for oil is an equivalent amount of apple sauce. It makes it more moist and healthier. I've heard that for people with juicers, the pulp can be used, but I'm not sure how much.

Most people do not know that bean puree can be substituted for ALL THE FAT! If you are new to this, you may want to experiment by only replacing 1/2 the fat with bean puree. Keep in mind that you never want to choose a bean that is darker in color than the cake you are making. For instance, if you are making a white cake; use white beans. If you are making a spice cake; use pinto beans. If you are making a chocolate cake; use black beans. Never use black beans for a white cake. To make the bean puree: you can either cook beans yourself as directed on the bag or use a can of beans, then dump the beans and some water in a blender or food processor and puree. You don't want it runny but add enough water until it looks pureed. If the recipe calls for 1 cup oil or butter, you can use 1 cup of bean puree OR use half oil/butter and 1/2 bean puree. Not only will this lighten the fat but it will also "health-i-fy" your recipe by upping the amount of protein and fiber AND save you money (have you seen the price of butter lately--whew!). I do this ALL THE TIME with my family and they still love the sweet treats.