Does the modern synthesis theory of evolution deny that acquired characteristics can be inherited?
Generally, yes, this is not part of theory of evolution as it is generally accepted by most of the scientific community. 

For a trait to be subject to natural selection over several generations, it needs to have heritability, meaning it needs to show up in offspring (or their offspring). Generally, what is called an "acquired trait" does not fit that criterion. What's passed on is the genetic material of the parents and that doesn't change when the parent acquires these traits. 

For example, bonsai trees are kept small by environmental conditions, as such, their height could be called an acquired trait. Their offspring does not naturally grow as a bonsai tree. Giraffes did not grow longer necks because they stretched them, but because those giraffe ancestors with longer necks (which just happens randomly through mutation - you have varying neck lengths in a population) could reach food more easily and could survive when those with shorter necks didn't. The giraffe example is a classic one that was proposed by Jean-Baptiste Lamarck, whose theory of evolution (called Lamarckism) was superceded by Darwin's. 

There are traits where acquired versus genetic traits are hard to differentiate - while a good athlete does not have children who are athletic because their parent did sports, they are often also good athletes. For one thing, the parent might be genetically "suited" for sports and has passed these genes on. Another reason is that they will probably train their children and want them to become more athletic. 

I know of one example where something that could be called an acquired trait is passed on to offspring - methylation patterns. Those are modifications of the genetic material that happen during an organism's lifespan and aren't modifications of the actual genetic code - they can be passed on to offspring. Since these patterns influence gene expression, they have an effect on the offspring. However, that's usually not what people mean when they talk about acquired traits. 

Lamarckian inheritance