This is a question that occurs constantly and indeed is one that will continue to occur until metal amalgam fillings that contain mercury are considered inappropriate for whatever reason. As ever, there is no concrete evidence that mercury amalgam fillings are unsafe from a medical health perspective, although there is good evidence that mercury amalgam fillings can create a reaction on the gum (in certain individuals) which is usually inflammatory in nature. Common sense would dictate that the dental profession would have a high rate of the symptoms that are put down to amalgam fillings if there is a genuine link with ill health and this has never been demonstrated.
Therefore replacement of old metal fillings on health grounds 'alone' are not indicated and one must consider the replacement material and the potential problems associated with these prior to considering the means of filling removal.
1. Tooth coloured composite materials of resin with a glass filler are good aesthetically and have reasonable strength and wear characteristics, however they leak monomer initially which is a proven carcinogen in laboratory conditions and they tend to flex and fail relatively quickly in large cavities.
2. Gold is inert and has excellent properties in the mouth but is unaesthetic and costly.
3. Porcelain is inert, aesthetic and has good strength when compressed, however it is brittle and can fracture and also requires the tooth to be cut down alot to create space for the new filling, it is also as expensive or more so than gold alone.
Removal of old fillings should ideally be carried out with a barrier technique called rubber dam to ensure the waste material is sucked away rather than being swallowed to make the process as pleasant as possible for the patient. Swallowing or inhaling the aerosol from the dental drill water supply has been shown to transitorily raise the levels of mercury in the urine when several fillings are removed but no longlasting effect has been demonstrated.
As you can tell from my answer, this is a subject that has been exhaustively investigated with no real negative outcome given for metal amalgam fillings, however they are very unaesthetic and if this bothers you, have them replaced with the most appropriate material. As an experienced and reasonably well qualified dentist, I would be personally sorry to see the demise of amalgam fillings as they have excellent strength and longevity and are an excellent base for crowns and bridges.
So I would be a little wary of the dental health professional or 'alternative' or 'complementary therapist' who suggests it is a good idea for your general well being and health to remove a mouthful of servicible metal fillings.