How to implement different class types within one variable?

Inheritance is the way to go:

public class MessageDataBase
{
    public string MessageID { get; set; }
    public string Date { get; set; }
    public bool IsSent { get; set; }
}
public class MessageData : MessageDataBase
{
    public string EmployeeInfo { get; set; }
}

public class MultiMessageData : MessageDataBase
{
    public byte MessageType { get; set; }
    public string MessageTypeString { get; set; }
    public string Platform { get; set; }
    public string EmployeeInfo { get; set; }
}

public class SocialData : MessageDataBase
{
    public bool IsReceived { get; set; }
    public byte MessageType { get; set; }
    public string MessageTypeString { get; set; }
    public string Message { get; set; }
    public string UserName { get; set; }
}

Use an interface, but you will have to make common changes in the //..do some work here section

public interface IMessageData
{
    //Example required function for all MessageData classes
    void SetDate(DateTime date);
    void SetIsSent(bool date);
}
public abstract class BaseMessageData : IMessageData
{
    public string MessageID { get; set; }
    public string Date { get; set; }
    public bool IsSent { get; set; }

    public BaseMessageData(string messageID)
    {
        MessageID=messageID;
    }
    public void SetDate(DateTime date)
    {
        Date = date.ToString("MM/dd/yyyy");
    }
    public void SetIsSent(bool sentStatus)
    {
        IsSent = sentStatus;
    }
}
public class MessageData : BaseMessageData, IMessageData
{

    public string EmployeeInfo { get; set; }

}

public class MultiMessageData : MessageData,IMessageData
{
    public byte MessageType { get; set; }
    public string MessageType { get; set; }

    public string Platform { get; set; }
}

public class SocialData : BaseMessageData, IMessageData
{
    public bool IsReceived { get; set; }

    public byte MessageType { get; set; }
    //Can't have to properties named the same. 
    //public string MessageType { get; set; }

    public string Message { get; set; }

    public string UserName { get; set; }
}

Then in your SendData

public void SendData()
{
    List<IMessageData> data;

    if (logic == 1)
    {
        data = new List<MessageData>();
    }
    else if (logic == 2)
    {
        data = new List<MultiMessageData>();
    }
    else
    {
        data = new List<SocialData>();
    }

    // some work here...
    foreach (var d in data)
    {
        //Only methods that will work are ones declared in the IMessageData interface
        d.SesDate(DateTime.Now);
        d.SetIsSent(true);
    }

    // if I do that then this Send method I will change it to be a Generic T maybe.
    _sendRepository.Send(data);
}

You can use my open source library, OneOf https://github.com/mcintyre321/OneOf

A OneOf is a special Type which can hold one value from a range of Types, like a Discriminated Union in F#. It provides an alternative to using inheritance/polymorphism.

You would write your method like this:

public void SendData()
{
    OneOf<List<MessageData>, List<MultiMessageData>, List<SocialData>> data = null;

    if(logic == 1){
        data = new List<MessageData>();
    } else if(logic == 2){
        data = new List<MultiMessageData>();
    }else {
        data = new List<SocialData>();
    }

    // some work here...


    _sendRepository.Send(data);
}

Or you could have var data = List<OneOf<MessageData, MultiMessageData, SocialData>>(); if you prefer, especially if you want to have different Types in the list.

install-package OneOf

It supports net451 and netstandard1.6