Hi  arvindasp,
You need to re-apply the jquery footable plugin after partial postback.
Refer below example.
HTML
<asp:ScriptManager runat="server" />
<asp:UpdatePanel runat="server">
    <ContentTemplate>
        <asp:Timer ID="Timer1" runat="server" OnTick="RefreshGridView" Interval="2000" />
        <asp:Label ID="LblLastUpdatedMsg" ForeColor="Red" Text="" runat="server" />
        <asp:GridView ID="GridView1" CssClass="footable" runat="server" AutoGenerateColumns="false" Style="max-width: 500px">
            <Columns>
                <asp:BoundField DataField="Id" HeaderText="Customer Id" />
                <asp:BoundField DataField="Name" HeaderText="Customer Name" />
                <asp:BoundField DataField="Country" HeaderText="Country" />
                <asp:BoundField DataField="Salary" HeaderText="Salary" />
            </Columns>
        </asp:GridView>
        <br />
        <asp:Button Text="Submit" runat="server" />
    </ContentTemplate>
</asp:UpdatePanel>
JavaScript
<link rel="stylesheet" type="text/css" href="https://cdnjs.cloudflare.com/ajax/libs/jquery-footable/0.1.0/css/footable.min.css" />
<script type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/jquery/3.7.1/jquery.min.js"></script>
<script type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/jquery-footable/0.1.0/js/footable.min.js"></script>
<script type="text/javascript">
    $(function () {
        $('[id*=GridView1]').footable();
    });
    //On UpdatePanel Refresh
    var prm = Sys.WebForms.PageRequestManager.getInstance();
    if (prm != null) {
        prm.add_endRequest(function (sender, e) {
            if (sender._postBackSettings.panelsToUpdate != null) {
                $('[id*=GridView1]').footable();
            }
        });
    };
</script>
Namespaces
C#
using System.Data;
VB.Net
Imports System.Data
Code
C#
protected void Page_Load(object sender, EventArgs e)
{
    if (!this.IsPostBack)
    {
        this.PopulateGridView();
    }
}
protected void RefreshGridView(object sender, EventArgs e)
{
    this.PopulateGridView();
    LblLastUpdatedMsg.Text = "Last Updated at " + DateTime.Now;
}
private void PopulateGridView()
{
    DataTable dt = new DataTable();
    dt.Columns.AddRange(new DataColumn[4] {
            new DataColumn("Id"),
            new DataColumn("Name"),
            new DataColumn("Country"),
            new DataColumn("Salary")
        });
    dt.Rows.Add(1, "John Hammond", "United States", 70000);
    dt.Rows.Add(2, "Mudassar Khan", "India", 40000);
    dt.Rows.Add(3, "Suzanne Mathews", "France", 30000);
    dt.Rows.Add(4, "Robert Schidner", "Russia", 50000);
    GridView1.DataSource = dt;
    GridView1.DataBind();
    //Attribute to show the Plus Minus Button.
    GridView1.HeaderRow.Cells[0].Attributes["data-class"] = "expand";
    //Attribute to hide column in Phone.
    GridView1.HeaderRow.Cells[2].Attributes["data-hide"] = "phone";
    GridView1.HeaderRow.Cells[3].Attributes["data-hide"] = "phone";
    //Adds THEAD and TBODY to GridView.
    GridView1.HeaderRow.TableSection = TableRowSection.TableHeader;
}
VB.Net
Protected Sub Page_Load(sender As Object, e As EventArgs) Handles Me.Load
    If Not Me.IsPostBack Then
        Me.PopulateGridView()
    End If
End Sub
Protected Sub RefreshGridView(sender As Object, e As EventArgs)
    Me.PopulateGridView()
    LblLastUpdatedMsg.Text = "Last Updated at " & DateTime.Now
End Sub
Private Sub PopulateGridView()
    Dim dt As New DataTable()
    dt.Columns.AddRange(New DataColumn(3) {
                        New DataColumn("Id"),
                        New DataColumn("Name"),
                        New DataColumn("Country"),
                        New DataColumn("Salary")})
    dt.Rows.Add(1, "John Hammond", "United States", 70000)
    dt.Rows.Add(2, "Mudassar Khan", "India", 40000)
    dt.Rows.Add(3, "Suzanne Mathews", "France", 30000)
    dt.Rows.Add(4, "Robert Schidner", "Russia", 50000)
    GridView1.DataSource = dt
    GridView1.DataBind()
    'Attribute to show the Plus Minus Button.
    GridView1.HeaderRow.Cells(0).Attributes("data-class") = "expand"
    'Attribute to hide column in Phone.
    GridView1.HeaderRow.Cells(2).Attributes("data-hide") = "phone"
    GridView1.HeaderRow.Cells(3).Attributes("data-hide") = "phone"
    'Adds THEAD and TBODY to GridView.
    GridView1.HeaderRow.TableSection = TableRowSection.TableHeader
End Sub