Most of my patients think tofu and soy protein are particularly heart healthy food choices. Since tofu contains significant calcium and protein, it is often viewed as a healthier alternative to dairy (which has inappropriately been labeled as heart unhealthy).
A huge growth in the use of soy protein occurred between 1996 and 2009 with annual sales of foods containing soy expanding from $1 billion to $4.5 billion. This appears to have been driven by a perception that soy is more healthful than other sources of protein (especially animal protein).
Much of the success of soy foods followed a 1999 decision by the FDA which approved a food-labeling health claim for soy protein for the prevention of coronary heart disease (CHD):

25 grams of soy protein a day, as part of a diet low in saturated fat and cholesterol, may reduce the risk of heart disease.

 Does soy deserve this designation? Should we be purposefully trying to consume more soy to lower our risk of heart disease?

Early studies, which compared consumption of 25 grams of soy protein versus control protein consumption, suggested a slight reduction in total and bad cholesterol levels. The problem with these studies is that a flawed surrogate marker (cholesterol or bad cholesterol) is being studied in place of the real disease (atherosclerosis and its associated complications, including heart attack and stroke). We now know that dietary interventions or drug therapies that lower cholesterol don’t necessarily reduce heart attacks or prolong life.
In 2000, the AHA published a document supporting the concept that 50 grams of soy protein per day would reduce heart disease risk .
However the AHA reversed this recommendation in a 2006 publication finding that

 In the majority of 22 randomized trials, isolated soy protein with isoflavones, as compared with milk or other proteins, decreased LDL cholesterol concentrations; the average effect was approximately 3%. This reduction is very small relative to the large amount of soy protein tested in these studies, averaging 50 g, about half the usual total daily protein intake. No significant effects on HDL cholesterol, triglycerides, lipoprotein(a), or blood pressure were evident. Among 19 studies of soy isoflavones, the average effect on LDL cholesterol and other lipid risk factors was nil. Soy protein and isoflavones have not been shown to lessen vasomotor symptoms of menopause, and results are mixed with regard to soy’s ability to slow postmenopausal bone loss. The efficacy and safety of soy isoflavones for preventing or treating cancer of the breast, endometrium, and prostate are not established; evidence from clinical trials is meager and cautionary with regard to a possible adverse effect. For this reason, use of isoflavone supplements in food or pills is not recommended. Thus, earlier research indicating that soy protein has clinically important favorable effects as compared with other proteins has not been confirmed. 

There is no scientific evidence that consuming soy protein lowers your risk of heart disease. There is no evidence that substituting soy protein for animal protein lowers your risk of heart disease. Certainly, if you like tofu (does anyone really like tofu?) and/or you have a philosophical desire to avoid meat and dairy consumption, tofu can provide a lot of the protein and calcium that you cannot get from eating only vegetables.

What does the searching the Internet tell us about tofu?

A Google search on the health benefits of tofu reveals stridently negative and positive (allegedly  evidence-based) articles (as is typical for everything in the world of nutrition). Medical News Today (“a leading health care internet publishing company,” which gets 9,000,00 views a month for unknown reasons), for example, has an overwhelmingly positive article written by a dietician which claims:

Countless studies have suggested that increasing consumption of plant-based foods like tofu, decreases the risk of obesity and overall mortality, diabetes, and heart disease and promotes a healthy complexion and hair, increased energy, and overall lower weight

The “Foundation for Integrative Medicine” (when you see the word “integrative” before the word “medicine,” substitute “unproven” and move to another website. This is a marker for quackery) cites similar claims, adding that regular tofu consumption reduces breast and lung cancer and osteoporosis.
None of these claims are supported in the medical literature.
On the anti-tofu side, we have this blog post from a chiropractor (chiropractors are usually big advocates of “integrative” medicine) who finds unfermented soy consumption to be the cause of myriad health problems including:

• Breast Cancer
• Brain damage
• Infant abnormalities
• Thyroid disorders
• Kidney stones
• Immune system impairment
• Severe, and potentially fatal food allergies
• Impaired fertility
• Danger during pregnancy and nursing

None of these claims are supported by the medical literature
You can also read about why soy “May be a health risk and environmental Nightmare” here. The majority of soy grown in the US comes from genetically modified plants from Monsanto which have had a gene inserted that allows them to resist Roundup. Consequently, farmers can spray all the Roundup they want on the plants.
Nobody knows if this is a health risk or not. Monsanto likes to make the case that the overall effects of RoundupReady soy, as they like to call it, are positive, whereas Mother Jones writes that soy is “Scarier Than You Think”.
My bottom line recommendation on soy is that, like all other foods, we should try to consume it in its least industrially processed form as part of a balanced diet of real foods.
There is no scientifically proven reason to  avoid it or seek it out.