According to Eastern Orthodoxy, does Matthew 17:20 apply to all people?

Yes, the message is for everyone. The Scriptures are there for everyone to read and draw lessons and guidelines towards salvation.

According to Eastern Orthodox teachings, there are multiple planes of reading Scripture. If you read this allegorically, then the message is that you should never renounce faith and stengthen in it, even when facing an insurmountable "mountain" of a problem. Even if you don't see a way out, God will show you a way.

However, I feel the focus of your question is on the literal, namely the God-given powers to perform big miracles such as physically moving mountains. This is also in accord with Eastern Orthodox teachings. There are many recorded instances of miracle working saints, that lived many centuries after the Apostolic age, performing all sorts of miracles in plain sight for many people to see. More on that below.

The question comes from, I believe, the feeling that such miracles are unheard of in more recent times, therefore this commandment must be applicable to the Apostles only. Regading this, the Eastern Orthodox teaching has the following to say:

• Miracles can be seen and identified as such only by the faithful. Jesus did plenty of miracles in front of both disciples and pharisees. The former saw good in them, strengthened their faith and comitted them to spreading the Lord's teachings, while the latter saw deceit, witchcraft and devil-work and their hearts remained petrified. Similarly, after the feeding of the 5000, Jesus tell the crowd that they follow Him not because they saw miracles, but because they ate bread (John 6:26). In other words, a full stomach was more gratifying than witnessing a miracle.

• Miracles, when worked by holy mean and women, are not performed on command, for entertainment or to elevate their status. One of the reasons of being gifted these powers by Holy Spirit is their great humbleness, and that humbleness would be destroyed if the result of the miracle would be the self gratification of the miracle worker. Miracles happen only when they aid the witnesses to progress on their path to salvation.

According to many Church Fathers, Christ was not, in fact, referring to the faith of the Apostles, but rather to the faith of the man who came to them when He gave this teaching.

Theophylact's commentary on the previous verses (Matthew 17:16-18):

Do you see how the man has shifted the blame for his own lack of faith upon the disciples, saying that they were too weak to heal? The Lord, therefore, is shaming him for accusing the disciples, saying "O faithless generation," that is, "It is not so much the fault of the weakness of the disciples as it is of your lack of faith, which, being great, has prevailed over the equal measure of their strength." He rebukes not only this man, but everyone who lacks faith, even the bystanders.^*

Thus, Christ's teaching was for all who heard Him, and not just the Apostles (to whom the teaching seems not to have even been directed).

Cyril of Alexandria (378-444) provides the same interpretation of the parallel passage in Luke (9:37-43), though the reference to the mustard seed appears later in a different context:

The father therefore of the demoniac was rude and uncourteous: for he did not simply ask the healing of the child, and in so doing crown the healer with praises, but, on the contrary, spake contemptuously of the disciples, and found fault with the grace given them. For I brought him, he says, to Thy disciples, and they could not cast it out. And yet it was owing to your own want of faith that the grace did not avail. Don’t you perceive that you yourself are the cause that the child was not delivered from his severe illness?

^{* The Explanation of the Gospel According to St. Matthew (tr. from Greek; Chrysostom Press, 2008), p.149}