I noticed that some websites invite me to do a particular kind of activity daily. For example, Duolingo would challenge me to learn a language for 10 days in a row and StackExchange would give me a gold badge if I kept visiting the website for 100 days. I am under the impression that it doesn't really motivate me because I prefer to work in bursts (for example, do a three-days work in a day then rest for two days) compared to keeping a daily routine. I feel that I could have gone away from Duolingo because I couldn't keep up with their suggested pace and I wonder if it actually works like this. If so, what would a better solution, one that motivates both me and the users that actually can do a chunk of a task daily?
This is an interesting question, but I think if you look at some other websites or apps, they will cater for both types of user behaviours. For example, they will provide a reward or badge for continuous login/visits, and they will also provide a reward or badge for a burst of activity (e.g. 10 posts in one day). So you can see in StackExchange that there are badges for these types of behaviours, and there are even badges for people who have been using the website for an extended period of time (probably to reward loyalty and long term involvement) as well as reward for people who have participated at a specific time (e.g. during moderator election).
Although you can also see this as a way for them to drive their users to what they believe is the optimal process for learning languages (in Duolingo's case anyway), where steady progress is the preferred approach to build on existing knowledge, rather than short bursts since you will tend to forget things when the memory is overloaded.
(How) do rewards work?
The connection between rewards and motivation is quite complex, and is more on the side of "rewards reduce motivation", despite the superficial appearance of increasing it. They induce you to do a task, while lowering your intrinsic motivation of it, making it less likely that you will pick up the task again another time, and also reduce your desire to do the task properly. For a more thorough explanation, there are popular science books like Punished by rewards by Kohn, as well as a sizeable body of academic literature. What you describe is one of the common side effects of rewards.
Gamification and motivation
In my observation, gamified sites end up with three groups of users (note a user can change his type over time):
Users "hooked" by the gamification. For them, the ratio of reward desirability to effort is favorable. This happens usually when the user unconsciously ties the badges (or rep points) to a powerful internal motivator such as "proof that I'm good at something". It is also more likely to happen if the task for which they get points has a low drudgery factor or was deliberately engineered to induce flow, such as most actual games (1).
Users who are not influenced by the gamification. They disregard all the badges and points and concentrate on the task at hand. Their interaction with the site depends on their intrinsic motivation only.
Users slightly to severely demotivated by the rewards. Of course, the gamification rewards are not the only contributor to their motivation, so even when their influence is negative, many users have enough total motivation to continue using the site. It is just a bit harder to do so, so they use it less, or do the tasks more mechanically and with less pleasure, or under more stress. The "more stress" part is the one you pointed out, in this case feeling that you have to invest time when it's not convenient to you just to keep up with the badge.
The explanation above stems from my general knowledge of the working of rewards, as well as observation. I am not aware of any quantitative studies which do this specifically for web site users. It would be very interesting to know what is the proportion of users whose motivation is helped by rewards (first group) to the ones harmed (third group), but I am not aware of such studies. But anecdotally, I'd say that most people do not internalize the badges of a random website they use as an important personal goal.
What can you do?
As for the "what can I do" part: As the website builder, there is not much you can do about it. You can stop pestering your users with so many rewards, reducing both the pull on group 1 and the push against group 3. You can also try to tie your rewards to the users' emotional needs - a good example of this is the recent Stack Overflow user profile redo, in which they added an estimation called "X people helped" - it is an inaccurate heuristic based on page views, but it sure feels good to see a number saying that I help people :)
Making the task more pleasant for the user is rarely an option. Assuming that you have built your site well, there is no way to do such a change without changing your complete business model. No matter what the author of a tax-filing software does, he cannot make the task itself fun.
If you are the user of a gamified site, the best option is to try to enter the second group - cultivate a mild disinterest for all these badges and focus on learning in Duolingo for learning's sake. Being in the third group is really a bummer, because not only do you achieve less of your goal, it is also less pleasant doing so. Being in the first group might make you achieve more of your goal, but it comes at a cost. We have enough stress in our lives and have to allocate our attention to a few worthy goals. Among all the gamified sites you sign up for, very few or none deserve becoming one of your primary goals - but being in the first group means you get hooked, exhibiting all the symptoms of a mild addiction, especially neglecting to do other, important things.
(1) A thing people don't get about gamification: What binds people to games is the combination of several factors, including simple goals. Introducing these goals, together with the metrics needed to track if they were reached, is what is widely known as "gamification". But if the other factors for a game are not in place, the result feels not at all gamelike.