What is the Orthodox teaching about touching Jesus' body?

The person may have identified as 'Orthodox' but their doctrine was not orthodox (and therefore, more explicitly not actually reflective of Eastern Orthodox doctrine*) as it denies the clear sense of:

36 As they were talking about these things, Jesus himself stood among them, and said to them, “Peace to you!” 37 But they were startled and frightened and thought they saw a spirit. 38 And he said to them, “Why are you troubled, and why do doubts arise in your hearts? 39 See my hands and my feet, that it is I myself. Touch me, and see. For a spirit does not have flesh and bones as you see that I have.” - Luke 24:36-39 ESV (emphasis added)

In order to harmonise the sense of John 20:17 with John 20:24, Luke 24:39 it is commonly theorised that there were (at least) two ascensions - one immediately subsequent to Mary's encounter and another 40 days later detailed in Luke 24:51 & Acts 1:9.

*As the Eastern Orthodox churches do not as a general rule produce doctrines that blatantly contradict scripture.

I have no references to these beliefs, other than I was raised Orthodox. I was taught that lots of people touched Jesus before the crucifixion. It is after that He did not allow people to touch him,

"Touch me not; for I am not yet ascended to my Father." (John 20:17)

The reason Thomas was allowed to touch Him is because Thomas was always the most doubtful of all the apostles. Even after all the other apostles saw Jesus and believed, and told Thomas,

he said to them, “Unless I see the nail marks in his hands and put my finger where the nails were, and put my hand into his side, I will not believe.” (John 20:24)

Therefore, Christ let Thomas touch Him so Thomas would believe.

"Put your finger here; see my hands. Reach out your hand and put it into my side. Stop doubting and believe.” (John 20:27)

Also, if no-one had touched Him, the heretics would claim that He was not real (therefore someone had to touch Him, and who better than Thomas, who had a hard believing by sight).

There's no scriptural support to say that only Thomas and/or certain others had some special permission to touch Jesus. However there are some great examples of people reaching out and touching with no "dangerous" outcomes:

A great example is how Jesus subtly rebukes Simon (Peter) for not greeting him with a kiss when he entered his house:

Luke 7:44-47 ESV Then turning toward the woman he said to Simon, “Do you see this woman? I entered your house; you gave me no water for my feet, but she has wet my feet with her tears and wiped them with her hair. You gave me no kiss, but from the time I came in she has not ceased to kiss my feet. You did not anoint my head with oil, but she has anointed my feet with ointment. Therefore I tell you, her sins, which are many, are forgiven—for she loved much. But he who is forgiven little, loves little.”

Regarding the "don't touch me" moment with Mary, Orthodox teaching takes it a different way:

John 20:17 ESV Jesus said to her, “Do not cling to me, for I have not yet ascended to the Father; but go to my brothers and say to them, ‘I am ascending to my Father and your Father, to my God and your God.’”

[notice the link in the book/chapter reference]

Mary was so excited to see Jesus her immediate reaction was to hug him--and who could blame her! Notice how Jesus gives a reason why Mary shouldn't cling to him: "for I have not yet ascended to the Father." Orthodox teaching would say that Jesus was basically calming her down and telling her she doesn't need to hold so much because he's not going anywhere. And that she should go soon and tell her brothers. Mary probably held to Jesus and thought to herself "I'm never letting go!" Jesus was merely calming her down and telling her there's still work to be done.

The incident with Thomas seems more obvious because Jesus straight up told him to touch his hands and side:

John 20:26-29 ESV Eight days later, his disciples were inside again, and Thomas was with them. Although the doors were locked, Jesus came and stood among them and said, “Peace be with you.” Then he said to Thomas, “Put your finger here, and see my hands; and put out your hand, and place it in my side. Do not disbelieve, but believe.” Thomas answered him, “My Lord and my God!” Jesus said to him, “Have you believed because you have seen me? Blessed are those who have not seen and yet have believed.”

Jesus didn't want Mary (and others) touching him as they had before the crucifixion, because he would no longer be with them as he had been. John Chrysostom explains this in his Homilies on the Gospel of St. John. [emphasis mine]

Some assert, that she asked for spiritual grace, because she had heard Him when with the disciples say, “If I go to the Father, ‘I will ask Him, and He shall give you another Comforter.’” But how could she who was not present with the disciples have heard this? Besides, such an imagination is far from the meaning here. And how should she ask, when He had not yet gone to the Father? What then is the sense? Methinks that she wished still to converse with Him as before, and that in her joy she perceived nothing great in Him, although He had become far more excellent in the Flesh.

To lead her therefore from this idea, and that she might speak to Him with much awe, (for neither with the disciples doth He henceforth appear so familiar as before,) He raiseth her thoughts, that she should give more reverent heed to Him. To have said, “Approach Me not as ye did before, for matters are not in the same state, nor shall I henceforth be with you in the same way,” would have been harsh and high-sounding; but the saying, “I am not yet ascended to the Father,” though not painful to hear, was the saying of One declaring the same thing.

For by saying, “I am not yet ascended,” He showeth that He hasteth and presseth thither; and that it was not meet that One about to depart thither, and no longer to converse with men, should be looked on with the same feelings as before. And the sequel shows that this is the case. “Go and say unto the brethren, that I go unto My Father, and your Father, unto My God and your God.”

With Thomas it was another matter. When Jesus let Thomas touch him, it was a form of rebuke for Thomas' disbelief. Chrysostom again:

Jesus again presenteth himself to them, and waiteth not to be requested by Thomas, nor to hear any such thing, but before he had spoken, Himself prevented him, and fulfilled his desire; showing that even when he spake those words to the disciples, He was present. For He used the same words, and in a manner conveying a sharp rebuke, and instruction for the future.

For having said, “Reach hither thy finger, and behold My hands; and reach hither thy hand, and thrust it into My side”; He added, “And be not faithless, but believing.” Seest thou that his doubt proceeded from unbelief? But it was before he had received the Spirit; after that, it was no longer so, but, for the future, they were perfected.

And not in this way only did Jesus rebuke him, but also by what follows; for when he, being fully satisfied, breathed again, and cried aloud, “My Lord, and my God,” He saith, “Because thou hast seen Me, thou hast believed; blessed are they who have not seen, and yet have believed.”