Studies have shown that sleep after learning improves recall; see for example Sleep after learning aids memory recall by Steffen Gais et al (2006). Other studies have shown that things you learn 30 minutes before going to bed are memorised better than things you learn 2 hours before going to bed. (While it makes sense to use Ankidroid before going to bed from the point of view of memorisation, exposing yourself to screens—laptop, smartphone, TV—shortly before going to bed will make it harder to fall asleep.)
I don't know if better performance means better memorisation. For example, Miranda Hitti, in How Sleep Helps Memory and Learning (WebMD, June 2005), describes the following experiment: 12 young adults got finger-tapping exercises. One lesson was in the morning, followed by a test in the evening (12 hours later); one lesson was in the evening, followed by a test the next morning (12 hours later). The results were better in the morning test, possibly because of the night's rest between the lesson and the test. Unfortunately, the article does not say whether performance was better during the morning lesson.
That learning is more efficient during the morning than in the afternoon was already reported by Hermann Ebbinghaus in the 1890s. In 1970, an experiment by Alan Baddeley and others (Memory and Time of Day) pointed in the same direction; short-term memory appears to work better in the morning.
There is also a study that found that Morning Lectures Are Better Retained Than Afternoon Ones. The lectures in this study were anatomy lectures, so I'm not sure that the results would translate well to spaced repetition for vocabulary. However, the study found that retention was better in all areas that were tested: obvious facts, fine facts, maneuver, and memory based on understanding.
A study from 2008 (Effect of the time-of-day of training on explicit memory) found that training (or learning) during the afternoon had a better effect on long-term memory than training in the morning. This effect was observed regardless of the learner's "chronotype" (i.e. whether they are a morning person, an evening person, ...) and regardless of whether the test took place in the morning or the afternoon.
Based on what I have found so far, I can't draw definitive conclusions yet, because the number of studies cited here is too small. What is needed here is a thorough meta-analysis or a literature review.
I have focused more on the question whether it is better to learn in the morning than in the afternoon/evening. That learning is less effective when you are tired is something I take for granted without consulting research. Nonetheless, according to an overview of "Neurocognitive Consequences of Sleep Deprivation" by Durmer and Dinges (2005), sleep deprivation affects cognitive performance in areas such as the following (emphasis added):
- Attention-intensive performance is unstable with increased errors of omission and commission
- Cognitive slowing occurs in subject-paced tasks, while time pressure increases cognitive errors
- Both short-term recall and working memory performances decline
- Reduced learning (acquisition) of cognitive tasks
- Performance requiring divergent thinking deteriorates
To the extent that tiredness has similar effects to sleep deprivation, these findings could explain why tiredness has a negative impact on vocabulary learning.
Since this question uses Ankidroid as an example, it is worth pointing out that reviewing a deck while tired will give you more repetitions of cards you forgot. When a card needs to repeated too often, Anki will mark it as a "leech", which is one way in which you will notice the effect of being tired. (Some people try to "unleech" cards.)