A low testosterone guy needs to be careful about his training program: study after study has shown that overtraining significantly reduces testosterone levels for 1-4 days afterward! And, by the way, it doesnï¿½t matter whether itï¿½s endurance or weight training ï¿½ the downward effect on testosterone levels is the same.
Then why do the body building and exercise magazines say that testosterone is increased with intense exercise? Many studies have shown that after intense exercise, testosterone levels are elevated for about an hour. Both weight lifting and endurance exercise like running and biking will produce this phenomenon. What the mags donï¿½t mention is that this effect is due almost entirely due to "hemoconcentration" and "decreased clearance" which is a polite way of telling you that the increase in testosterone is NOT coming from your testes (or adrenal glands).
Now it is important to note that Weight Training Can Boost Testosterone in the long term if done correctly. In addition, there are other GREAT reasons to exercise, such as dramatically improving your erectile strength and rebuilding your brain. For details, see this Sexercise link.
However, many of us males have to admit that we often overdo things and overtraining is a prime example. And what many men do not realize is that some research shows that overtraining can do the following:
a) Lower testosterone
b) Raise cortisol
c) Decrease the testosterone-to-cortisol ratio
Now not all studies show this, but some do and let me cover a few of them below:
1. Rugby Players. One study on rugby players took testosterone reads and compared them with questionnaires filled out by the players. What they found was that overtraining synptoms correlated nicely with total testosterone levels. How much did overtraining lower testosterone in these athletes? Well, if you look at the data points, about a fourth of the players that were significantly into the overtraining zone had drops in testosterone of 20-30%.
However, the most important stats were probably in an athlete on the very end of the spectrum, whose questionairre showed him to be the most overtrained. His testosterone was about half of his previous level, a very significant drop obviously. The authors concluded that "the questionnaire may be a useful tool for screening subjects at risk of overtraining. Testosterone concentration is influenced by tiredness, and is therefore a valid marker of tiredness." 
2. Elite Basketball Players. One study found that there was an actual cutoff in game playing time that could generally be associated with hornonal disruptions:
"Players who played between 13 and 25 minutes per game showed the highest values of TT (22.8 ï¿½ 6.9 nmolï¿½L-1) and TT/C (47.1 ï¿½ 21.2). March and April showed the most catabolic or stressed hormonal state (low TT/C values and high ones of cortisol) and that is necessary to take into account according to PT (>25-minute per game) and specific playing position. Monitoring plasma TT and cortisol is recommended to prevent excessive stress caused by professional basketball season requirements." 
3. Elite Football Players. This also used a questionnaire with elite athletes and found that morning testosterone was associated with the training syndrome:
"The results of data analyses showed that the overtraining score from questionnaire correlates with cortisol concentration at 8,11 am and mean cortisol concentration on rest day respectively (r=0.71, r=0.62, r=0.61; pï¿½0.01), testosterone concentration at 8 am (r=0.42; p=0.05) and testosterone/cortisol ratio at 8 am (r=-0.42; pï¿½0.05). Result show that, the questionnaire may be a useful tool for monitoring and preventing of overtraining syndrome." 
NOTE: If you know someone who may be overtraining, check out my page on The Best Signs of Overtraining.
Several recent studies have come out showing that overly intense exercise whacks your bodies ability to fight colds and infections.  This is bad for many reason, but especially because it is self-defeating. One cold or flu can keep us out of the gym for a week or longer.
One study on rats found that overexercise slammed immunoglobulin-A.  And another researcher pointed out that overtraining can lead to "reduced catecholamine excretion, frequent illness, disturbed sleep and alterations in mood state... decreases in neutrophil function, serum and salivary immunoglobulin concentrations and natural killer cell number and possibly cytotoxic activity in peripheral blood".  Some have even speculated that chronic overtraining and the ensuing compromised immune system could lead to cancer. This is nasty, scary stuff!
Overtraining will, by the way, likely whack your sperm count while it's at it. Spanish researchers found that sperm levels went down by 50% on young, healthy cyclers.  (It took three days for sperm levels to return to normal levels.) No telling what it does to us "more mature" guys!
Intense training has even found to effect the liver in healthy men! Researchers found abnormally elevated levels of many critical liver proteins.  Of course, that's not to say that intense exercise will kill you, but it does point out that intense exercise stresses the body in ways that we do not even understand yet. I also mention it because if you are on HRT, they should test liver function. If you get an abnormal reading, you may want to go easy on the workouts and get retested. They recently found that, at least in women, moderate levels of exercise were helpful for long term memory and cognition, but strenuous, long term exercise (such as marathons) were actually damaging to long term memory!  Of course, it could be some unknown factor in these women's lifestyle, but the odds are that elevated cortisol is wreaking its usual havoc on the brain in this case.
If you want to do intense workouts, I highly recommend SLOWLY building up. Personally, I have found that the key (for me) is to start with a moderate workout and then slowly increase the intensity as the weeks go by. This gradual pattern does not seem to disturb my testosterone levels or libido.
ATHLETES: Overtraining can seriously affect performance. Researchers have identified several short term key markers that result from overtraining, including "impaired anaerobic lactic acid perforance and a reduced time to exhaustion".  The same study discusses many other soon-to-follow issues including problems with uric acid levels, ammonia, creatine kinase, free testosterone/cortisol, growth hormone and so on. More subjective issues include excessively sore muscles and sleep and mood disorders. Again, overtraining is real and leads to real physiological issues.
1) Jour Appl Physio, 2007, 103:693-99
2) JAMA. 2006;296:2307
3) British J Clin Pharm,2008, 65:253-259
4) Alzheimer's Association 2009 International Conference on Alzheimer's Disease, Vienna, Austria, July 11-16, 2009. Mary C. Tierney
5) Scandanavian J Med Sci Sports, 2008, 18:367-372
6) Immunology and Cell Biology (2000) 78, 502ï¿½509; "Overtraining effects on immunity and performance in athletes", Laurel T MacKinnon
7) Sports Medicine, Feb 1 2002, 32(2):95-102, "Diagnosis of Overtraining: What Tools do We Have?"
8) Journal of Strength & Conditioning Research, February 2015, 29(2):368ï¿½378, "Using Testosterone and Cortisol as Biomarker for Training Individualization in Elite Basketball: A 4-Year Follow-up Study"
9) FALL 2009 , Volume 17 , 3(47):127 To 137, "Paper: THE RELATIONSHIP BETWEEN SALIVARY TESTOSTERONE AND CORTISOL CONCENTRATIONS WITH PSYCHOLOGICAL OVERTRAINING SYMPTOMS IN ELITE FOOTBALL PLAYERS"
10) Br J Sports Med, 2004, 38:260-263, "Salivary testosterone and cortisol in rugby players: correlation with psychological overtraining items"