Is there a term that describes the personality trait of always desiring more? I see this kind of trait in some of the people who fear the commitment (in a marriage) since they don't want to have to renounce their vows when their feelings toward their spouses change.

I see this kind of trait also in those who go to discos to find a better partner, who looks for the perfection, who thinks they deserve more, they seem unable to be satisfied with what they have. It happens often with narcissism. I think it's a way to manage anxiety: "I'm not enough, I need external validation, I must be more and more to be sure I will always be loved, because I'm not good.. and the other will realize.. so he must love me really a lot". 

Some people prefer not to build a good relationship because they are scared to lose the chance for their better relationship. Some expect to get very strong feelings and emotions from a relationship, and that they will find someone who is able to really "get" them.

I would call it the illusion of being able to fully satisfy our expectations, without the ability to renounce the things they have, or to be easily satisfied by them.

What could this phenomenon be named and related to?
I'm not entirely sure what you mean by "renounce," but you may find some answers to your question in the fields of social psychology and behavioral economics. The first part of your question describes someone who is unwilling to commit or make a decision for fear of losing out on future opportunities. There has been research to indicate that having more options when making a decision is actually stressful and can provoke anxiety (Ackerman, Gross, &amp; Celly, 2014). 

One recent study found that individuals who were presented with all options at once were more satisfied than those who were shown options one at at time (Mogilner, Shiv, &amp; Iyengar, 2013). The individuals who were given options one at a time tended to imagine a better, ideal option was yet to come; the authors noted that these participants were less happy with their choices in part because of that hope. The introduction to this paper references several other papers which found that individuals who were focused on possible options were less satisfied with the choices they may and were less likely to stick with those choices because of the hope for "something better." The paper specifically cites several articles which indicate that a person's commitment to their romantic partner is not necessarily influenced by the partner's characteristics, but by the person's focus on other options (Gange´ and Lydon 2001; Johnson and Rusbult 1989; Lydon et al. 1999; Rusbult, Martz, and Agnew 1998). Similarly, Miller (1997) found that awareness of and attention to alternative partners was the highest predictor of failure in romantic relationships. 

How people approach decisions also impacts how they address the options presented. In choice deferral, a person puts off a decision that requires difficult compromises (such as committing to one partner and thus losing the freedom to continue dating) in order to avoid the negative emotions linked to that decision (Luce, 1998). Some people may be more prone to this behavior than others. Schwartz and colleagues (2002) have proposed that people fall into two groups: those who seek to maximize their decision by finding the very best option out of choices presented (maximizers), and those who have an idea in mind of what would be an "acceptable" option and settle for the first choice that surpasses their expectations (satisficers). Maximizers are always on the look out for "the best," are less satisfied with their choices, more likely to "socially compare" their options to others, and more likely to experience regret (Ma &amp; Roese, 2014). Another recent publication proposed that when individuals experience "choice closure," and stop re-evaluating the non-selected alternative options, satisfaction with choice can increase (Gu, Botti, &amp; Farro, 2013). 

Again, I am not certain I understood all of your question, but I hope that this provides you with a starting point to find the answer. 

Ackerman, D., Gross, B. L., &amp; Celly, K. S. (2014). Having many choice options seems like a great idea, but...: Student perceptions about the level of choice for a project topic in a marketing course. Journal of Marketing Education. April, e-pub ahead of print. Doi: 10.1177/0273475314522038 

Gagne, F. M., &amp; Lydon, J. E. (2001). Mindset and relationship illusions: The moderating effects of domain specificity and relationship commitment. Personality and Social Psychology Bulletin, 27(9), 1144-1155. 

Gu, Y., Botti, S., &amp; Faro, D. (2013). Turning the page: The impact of choice closure on satisfaction. Journal of Consumer Research, 40(2), 268-283. DOI: 10.1086/670252

Johnson, D. J., &amp; Rusbult, C. E. (1989). Resisting temptation: Devaluation of alternative partners as a means of maintaining commitment in close relationships. Journal of Personality and Social Psychology, 57(6), 967-980.

Luce, M. F. (1998). Choosing to avoid: Coping with negatively emotion-laden consumer decisions. Journal of Consumer Research, 24, 409-433. 

Ma, J., &amp; Roese, N. T. (2014). The maximizing mind-set. Journal of Consumer Research, 41(1), 71-92. DOI: 10.1086/674977

Miller, R. S. (1997). Inattentive and contented: Relationship commitment and attention to alternatives. Journal of Personality and Social Psychology, 37(4), 758-766. 

Mogilner, C., Shiv, B., &amp; Iyengar, S. S. (2013). Eternal quest for the best: Sequential (vs. simultaneous) option presentation undermines choice commitment. Journal Of Consumer Research, 39(6), 1300-1312. doi:10.1086/668534

Rusbult, C. E., Martz, J. M., &amp; Agnew, C. R. (1998). The Investment Model Scale: Measuring commitment level, satisfaction level, quality of alternatives, and investment size. Personal Relationships, 5(4), 357-391.  

Schwartz, B., Ward, A., Monterosso, J., Lyubomirsky, S., White, K., &amp; Lehman, D. R. (2002). Maximizing versus satisficing: Happiness is a matter of choice. Journal Of Personality And Social Psychology, 83(5), 1178-1197. doi:10.1037/0022-3514.83.5.1178