This is a very popular question in dentistry today; one which does not have a simple answer.
Silver fillings are composed of a variety of metals including: mercury, silver, tin, and copper. Composite fillings are made of a combination of dimethacrylate monomers(acrylic) and silica fillers (glass). Both options have their indications and limitations depending on the circumstance.
Amalgam fillings have been around since the early 1800's. It's quite amazing that for the most part it's composition is relatively unchanged. The most controversial aspect of the amalgam filling is the mercury component. Even though there is no scientific consensus to date linking any ill-health effect to the mercury release in dental amalgam restorations, some individuals choose to remove these fillings and replace them with non-amalgam like materials.
Generally, removing a sound filling for the sole purpose of replacing it with a white filling is not recommended.
Any time a tooth is drilled on certain inherent risks need to be understood. Irritation to the tooth can create sensitivity(transient or long term which can potentially lead to the need for root canal treatment). Small craze lines can develop in teeth which overtime can lead to crack formation. This may then require crown treatment, which again may create a sensitive tooth!
However, if a situation exists like decay or a poor seal around an amalgam, then replacing it with a white filling is at this time possible.
Placing a white filling for the first time on a tooth is preferable(especially if it is small) because the amount of tooth structure removed will be less than it would be for a silver filling. It is the bonding of the composite to tooth structure that allows a more conservative drilling approach. Silver fillings are ‘wedged' into a space in the tooth that is created by drilling. This space has to be of a certain size for the material to be durable; this space is usually larger than for a composite restoration.
If you have a concern regarding the amalgam filings in your mouth it is best to speak with your dentist to see what the best approach would be for you.