Praise be to Allah.
Undoubtedly clothing comes under the heading of valuable wealth that should be protected against damage and waste, and Islamic teachings warn against wasting it in ways that are not beneficial.
Allah, may He be exalted, says (interpretation of the meaning):
{O children of Adam, take your adornment at every masjid, and eat and drink, but be not excessive. Indeed, He likes not those who commit excess} [al-A‘raaf 7:31]
{and do not spend wastefully.
Indeed, the wasteful are brothers of the devils, and ever has Satan been to his Lord ungrateful} [al-Isra’ 17:26-27].
It was narrated from al-Mugheerah ibn Shu‘bah: I heard the Prophet (blessings and peace of Allah be upon him) say: “Allah hates three things for you: gossip, wasting money, and asking too many questions.” Narrated by al-Bukhaari (1477) and Muslim (593).
Al-Ameer as-San‘aani (may Allah have mercy on him) said:
What is meant by Allah’s hating a thing is that He is not pleased with it and does not like it.
But does that indicate that it is prohibited, when it is narrated that He hates it? That is possible." (At-Tanweer 3/327).
Undoubtedly signing shirts means that they can no longer be used for wearing and will be wastedmoreover, there is no benefit or purpose worth mentioning in signing them. Doing this is a kind of wasting money and being extravagant with it.
If we assume that there was some benefit in the signatures, that could be achieved by signing in a diary that is specifically for that purpose, or signing notebooks and the like, that does not spoil the property.
What the Muslim should do in his daily conduct is weigh up the pros and cons in the light of Islamic teachings. If the negative outcomes (cons) outweigh the positive outcomes (pros), then he should avoid it. This is what is dictated by prudence.
Shaykh al-Islam Ibn Taymiyah (may Allah have mercy on him) said:
Perfect prudence means knowing the better of two good things and the worse of two evils, and he should be aware that Islamic teachings are aimed at attaining and perfecting good things, and eliminating and reducing bad things. Otherwise, the one who does not weigh up pros and cons in the light of Islamic teachings may end up refraining from doing things that are obligatory and doing things that are prohibited." (Majmoo‘ al-Fataawa 10/512).
Conclusion:
This practice comes under the heading of wasting money and extravagance that is not allowed, because it serves no significant purpose. If we assume that it does serve some purpose, that purpose may be achieved in a different way, without wasting money and being extravagant.
And Allah knows best.