Q:

Does the color temperature of a computer screen affect sleep patterns?

There's a little tool called f.lux that claims:

During the day, computer screens look good — they're designed to look like the sun. But, at 9PM, 10PM, or 3AM, you probably shouldn't be looking at the sun.
F.lux fixes this: it makes the color of your computer's display adapt to the time of day, warm at night and like sunlight during the day.
It's even possible that you're staying up too late because of your computer. You could use f.lux because it makes you sleep better, or you could just use it just because it makes your computer look better.

Is it true that the color temperature of a computer screen could upset one's biorhythm?

A:

One study says, "possibly".  Note that the illumination provided by the light sources used in the study may not be consistent with the light coming from a computer monitor.  Also note that the sample size is not large, and a mechanism for the effect is not proposed.  (source)

J Physiol Anthropol Appl Human Sci. 2005 Mar;24(2):183-6.
Effect of color temperature of light sources on slow-wave sleep.
Kozaki T, Kitamura S, Higashihara Y, Ishibashi K, Noguchi H, Yasukouchi A.
Department of Physiological Anthropology, Faculty of Design, Kyushu University, Japan. kozaki@design.kyushu-u.ac.jp
In order to examine whether the spectral compositions of light source may affect sleep quality, sleep architecture under different color temperatures of light sources was evaluated. Seven healthy males were exposed to the light sources of different color temperatures (3000 K, 5000 K and 6700 K) for 6.5 h before sleep. The horizontal illuminance level was kept at 1000 lux. Subjects slept on a bed in near darkness (< 10 lux) after extinguishing the light, and polysomnograms recorded the sleep parameters. In the early phase of the sleep period, the amount of stage-4 sleep (S4-sleep) was significantly attenuated under the higher color temperature of 6700 K compared with the lower color temperature of 3000 K. Present findings suggest that light sources with higher color temperatures may affect sleep quality in a view that S4-sleep period is important for sleep quality.

A recent article in Scientific American (source) discussed a possible mechanism for the effect.

Many years later researchers extended Keeler’s observation, showing that mice genetically engineered to lack rods and cones (the light receptors involved in vision) nonetheless reacted to changes in light by adjusting their circadian clock—the internal timer that synchronizes hormone activity, body temperature and sleep. The animals performed the usual daytime activities when in daylight and nighttime activities when in the dark. They could do so even though their retinas lacked the photoreceptor cells that vertebrate eyes use to form images, although surgically removing their eyes abolished this ability. This phenomenon may be common to many mammals, including humans: recent experiments have shown that certain blind people can also adjust their circadian clocks and constrict their pupils in response to light.

A:

Yes, the blue colour may affect sleep cycles, but an app might not be enough to fix it, depending on other lighting conditions.
The existing answers are somewhat theoretical, without considering whether there is a pragmatic difference caused by real computer screens.
There has been a lot of research into the effect of blue light from computer screens and smart-phones and how they affect circadian rhythyms in humans.
Does screen light have any effect at all?

Evening use of light-emitting eReaders negatively affects sleep, circadian timing, and next-morning alertness, Anne-Marie Chang, Daniel Aeschbach, Jeanne F. Duffy, and Charles A. Czeisler
PNAS January 27, 2015 112 (4) 1232-1237 DOI: 10.1073/pnas.1418490112

Twelve young adults were put in a dim room for four hours before bedtime, for five nights in a row. Each night, half read a printed book and half read from an eBook which emitted light.

We found that the use of these devices before bedtime prolongs the time it takes to fall asleep, delays the circadian clock, suppresses levels of the sleep-promoting hormone melatonin, reduces the amount and delays the timing of REM sleep, and reduces alertness the following morning. Use of light-emitting devices immediately before bedtime also increases alertness at that time, which may lead users to delay bedtime at home. Overall, we found that the use of portable light-emitting devices immediately before bedtime has biological effects that may perpetuate sleep deficiency and disrupt circadian rhythms, both of which can have adverse impacts on performance, health, and safety.

It is worth noting that a letter in response argued that the effect was confounded by the subjects not being exposed to normal light patterns during the day.

That is, the light emitted from the eReaders would have had a much smaller effect in alerting the brain than they would have had the participants been exposed to a normal pattern of everyday light exposure before using the eReaders before bedtime. Thus, the question still remains as to whether the light being emitted from an eReader, or any other type of electronic device, would actually impact nocturnal alertness and sleep in normally behaving individuals.

However, the original authors responded rejecting that their study had this shortcoming:

However, we disagree with the inference that the lighting conditions in our laboratory study overestimated the real-life effects of reading from a light-emitting eReader compared with reading from a printed book. First, participants in these studies were not “spending the entire day in dim room lighting.” [...] Second, such minor differences in daytime ambient lighting are unlikely to account for our findings.

(If it is easier reading, this study was discussed and put in context in this essay.
Is the Colour-Temperature Relevant?

Light level and duration of exposure determine the impact of self-luminous tablets on melatonin suppression, Brittany Wood, Mark S.Rea, Barbara Plitnick, Mariana G.Figueiro, Applied Ergonomics
Volume 44, Issue 2, March 2013, Pages 237-240, DOI: 10.1016/j.apergo.2012.07.008

In this study thirteen participants looked at iPad screens for a couple of hours under different conditions - just the screen (on full brightness), with additional blue leds shining on them, and with orange-tinted goggles that cut out the blue light. They had they melatonin-levels measured.

The present study extends results from Figueiro et al. (2011)
showing that a 2-h exposure to self-luminous tablets can result in a measurable, statistically reliable suppression of melatonin in Table 1.

That is, not wearing the orange goggles to cut out the blue light, lead to measurable changes in their melatonin levels, when viewing a regular iPad screen. Adding additional blue light increased the effect.
They add:

it is important to acknowledge that usage of self-luminous electronic devices before sleep may disrupt sleep even
if melatonin is not suppressed. Clearly, the tasks themselves may be
alerting or stressful stimuli that can lead to sleep disruption. For
now, however, it is recommended that these devices be dimmed at
night as much as possible in order to minimize melatonin
suppression, and that the duration of use be limited prior to
bedtimes.

What can be done?

Analysis of circadian properties and healthy levels of blue light from smartphones at night, Ji Hye Oh, Heeyeon Yoo, Hoo Keun Park & Young Rag Do, Scientific Reports volume 5, Article number: 11325 (2015)

This article takes a relatively theoretical approach, estimating the amount of different types of light that are produced by smartphone displays in typical use, and proposes some approaches to minimise the effects:

At night, if we use smartphone displays that are darker than that of the CIL threshold of light which activates circadian systems, the unhealthy effect of smartphone displays can be minimized.

It still requires further experimental support. However, they were able to highlight the limits of what can be achieved by adjusting the smartphone alone.

If we use smartphones at the proper distance and brightness setting in a dark room, the MSV can drop below ~1%. However, the use of a smartphone in a bright room at night significantly increases the [circadian illuminance] and MSV values [...] If people use smartphones in a bright room at night, it will be a little difficult to decrease the blue effect of smartphones on human health by varying smartphone variables that users can adjust. From the results of tuning the emitting wavelength of blue LEDs in a smartphone LCD backlight, fine control of blue light in smartphone displays can have a greater impact on reducing the unhealthy effect of blue light from smartphone displays at night. Thus, combating the unhealthy effect of blue light from smartphone displays at night does not simply mean turning off the smartphone, but rather using well designed [spectral power distribution] and the proper intensity of smartphone displays to see better where and when required while protecting people’s health and circadian rhythm.

