Can a priest resign? Is a priest permitted to resign from his position? By this I mean, can he voluntarily choose to stop serving as a priest, and no longer be under the restrictions of a priest (e.g. that they cannot marry)?
The answer is both 'yes' and 'no'.  A priest can resign from the priesthood and seek dispensation from his clerical obligations, yet the Church will always regard him as a 'laicised' priest, since ordination is regarded as irrevocable.

Canon 1583: It is true that someone validly ordained can, for a just reason, be discharged from the obligations and functions linked to ordination, or can be forbidden to exercise them; but he cannot become a layman again in the strict sense, because the character imprinted by ordination is for ever. the vocation and mission received on the day of his ordination mark him permanently.
Canon 291: Apart from the case mentioned in can. 290, n. 1, loss of the clerical state does not entail a dispensation from the obligation of celibacy, which only the Roman Pontiff grants.
Canon 292: A cleric who loses the clerical state according to the norm of law loses with it the rights proper to the clerical state and is no longer bound by any obligations of the clerical state, without prejudice to the prescript of can. 291. He is prohibited from exercising the power of orders, without prejudice to the prescript of can. 976. By the loss of the clerical state, he is deprived of all offices, functions, and any delegated power.
Canon 293: A cleric who loses the clerical state cannot be enrolled among clerics again except through a rescript of the Apostolic See.

GianPaolo Salvini, S.J. says, in 'Priests Who &quot;Desert,&quot; Priests Who &quot;Come Back&quot; ', Of the 1,076 priests who leave the ministry each year, 554 ask for a dispensation from the obligations imposed by the priestly state: celibacy, and the recitation of the breviary. Salvini estimates the number of former priests who became married as in the thousands, but concedes that no statistic is available.
This Press Statement was issued online by John Wijngaards in 1998, giving his reasons for resignation from ministry as &quot;Catholic Priest Resigns Due To Conflict Of Conscience&quot;.
This Letter of Resignation by Athanasio Dzadagu, giving his reason as the wish to marry, was reproduced by Zim Catholic news in its blog. The letter received both favourable and unfavourable comments.