What are the effects if boiler is operating below its rated capacity?

It depends on how much under the rated capacity you want to operate the boiler. Different boilers can have a different and wide range of turndown and operating ranges. The biggest concern you need to have is about the boiler water circulation. As you change the operating pressure, the specific volume (density) changes. When a boiler is running under rated load/capacity then the main effect is the decreased efficiency of boiler. Other effects depend upon the type of boiler, type of fuel used, and type of water circulation.

Typical turndown on a gas, oil or coal burner can range from 10:1 for gas to 4:1 for coal depending in the burner. Do not confuse burner turn down with operating the boiler at a lower pressure. Yes the burners can be modulated such that the load from the boiler will be less. A boiler is rated for different MCRs. Typically the once through boilers are rated at 70%, 95% and 100% MCR. Typically you do not want to run the boiler less than 25% of MCR steam load 4:1 turndown) - you have to ensure superheater is kept cool with adequate steam flow. You can operate the boiler at a lower steam flow based on adequate cooling through the circuits to ensure cooling.

Modern boilers are equipped with Modulating burner. Boilers go to the low firing position according to load requirement. If burner is on off type they fire according load firing position or if burner is feedback controller type the adjust according to load according condition. 

However operating large capacity boiler at low firing position degrades the efficiency and exact characteristic curve can be obtained from supplier.