My Mother and aunties always asked with alarm how I got even a minor scratch when I was young. If an injury was caused by anything rusty, I was to be transported immediately to the doctor for a "tetanus shot" so I would not get "lockjaw". Do injuries from rusty object cause tetanus? Is there a significant chance of contracting tetanus this way in first world countries?
Rusty or not, any contaminated object that causes an injury could result the injured person to have a Tetanus infection. And it is not just the first world countries, because Tetanus occurs worldwide.
More than a rusty nail
The New York Times has demystified the claim in a short article , which is good for a prompt reading. It is more than just a rusty nail for one to have a Tetanus infection.
A rusty nail will do. But the infection can come from many sources -- sewing needles, animal bites, gardening tools, splinters. Injuries that create dead skin, like burns and frostbite, can also lead to infection.
Quora has similarly short answers  to debunk the claim; The most upvoted answer explained well in a paragraph, albeit had misspelt the bacteria name (corrected name with emphasis mine).
It is neither the nail nor the rust that causes Tetanus.
Tetanus is caused by the infection with Clostridium Tetani, an organism that is ubiquitous and more likely to be found in dust and dirty places. If it is present on a rusty nail, it can cause Tetanus, only if the person is not immunized against it.
HowStuffWorks has explained further on the exposure of the bacteria to the people. Neither people in farms and cities are completely safe, according to its article , notably page 2 as quoted below.
Tetanus is caused by bacteria known as Clostridium tetani, which is commonly found in soil, dust and animal feces. Because of its presence in soil and manure, gardeners and others who work in agriculture are particularly at risk for exposure to this bacteria [...] But city-dwellers aren't completely safe -- a dusty sidewalk or street may harbor just as many bacteria.
Clostridium tetani (C. tetani)
Centers for Disease Control and Precention (CDC) explains about Tetanus and the bacteria that causes the infection in its publication, which is available as an HTML page  and a PDF file .
Tetanus is an acute, often fatal, disease caused by an exotoxin produced by the bacterium Clostridium tetani. [...]
The organism (C. tetani) is sensitive to heat and cannot survive in the presence of oxygen. The spores, in contrast, are very resistant to heat and the usual antiseptics. They can survive autoclaving at 249.8°F (121°C) for 10-15 minutes. The spores are also relatively resistant to phenol and other chemical agents.
There is no mention of "rust" found in the text of publication, which would disagree with the claim that "rusty nail wounds cause tetanus". Regardless of minor or major wounds, contamination leads to a successful infection.
Transmission is primarily by contaminated wounds (apparent and inapparent). The wound may be major or minor. In recent years, however, a higher proportion of patients had minor wounds, probably because severe wounds are more likely to be properly managed. Tetanus may follow elective surgery, burns, deep puncture wounds, crush wounds, otitis media (ear infections), dental infection, animal bites, abortion, and pregnancy.
Tetanus treatment and prevention
While Tetanus is a preventable disease , a person who has been infected should be treated immediately with human tetanus immune globulin (TIG) (or equine antitoxin) .
Tetanus is not contagious from person to person. It is the only vaccine-preventable disease that is infectious but not contagious.
One should have received Tetanus vaccine when one was younger than 7 years of age. A booster dose may be given to adults, but the amount of dose varies by whether the adult was unvaccinated or after an exposure to the infection under some circumstances .
 The Claim: Stepping on a Rusty Nail Can Cause Tetanus by Anahad O'Connor, Feb. 22, 2005.
 If you step on a rusty nail, will you really get tetanus? on Quora, asked 15 weeks ago to this date.
 If you step on a rusty nail, will you really get tetanus? on HowStuffWorks.
 Tetanus Chapter of Pinkbook: Epidemiology and Prevention of Vaccine-Preventable Diseases on CDC, 13th Edition (2015) retrieved on Nov. 24, 2015.
 Printer friendly version of Tetanus Chapter of Pinkbook: Epidemiology and Prevention of Vaccine-Preventable Diseases on CDC, 13th Edition (2015) retrieved on Nov. 24, 2015.
 Tetanus: Diagnosis and Treatment on CDC.
 Tetanus Immune Globulin (Injection) on National Library of Medicine - PubMed Health.
 Vaccines: VPD-VAC/Tetanus/main page on CDC.
Tetanus toxin, which has been given the scientific name tetanospasmin, is created when the bacterium Clostridium tetani divides and grows. C. tetani cannot divide in an oxygenated environment, but as a piece of iron flakes off and rusts, it robs the surrounding environment of oxygen. This is the easiest way for C. tetani to reproduce and create toxins (source 1). It is the exposure of an iron surface, not the amount of rust, which makes it possible for C. tetani to become potent, but rust is obviously a danger sign.
Tetanus toxin is one of the most potent neurotoxins in the world. If your skin is broken by contact with rusting iron and active C. tetani spores enter the bloodstream, the toxin will start work immediately and it is indeed worth a hospital visit. Lockjaw, caused by the toxin binding itself to more vulnerable parts of the nervous system, has been known as one of the first signs of tetanus poisoning since ancient times (source 2).
Demain AL, Gerson DF, Kole M, Fang A. "The role of reduced iron powder in the fermentative production of tetanus toxin." Appl Microbiol Biotechnol. 2006 Nov;73(1):55-9. Epub 2006 Apr 19.
J M Pearce. "Notes on tetanus (lockjaw)." J Neurol Neurosurg Psychiatry. 1996 Mar; 60(3): 332.