Most solutions with awk leave an space. The options here avoid that problem.
Option 1
A simple cut solution (works only with single delimiters):
command | cut -d' ' -f3-
Option 2
Forcing an awk re-calc sometimes remove the added leading space (OFS) left by removing the first fields (works with some versions of awk):
command | awk '{ $1=$2="";$0=$0;} NF=NF'
Option 3
Printing each field formatted with printf will give more control:
$ in='    1    2  3     4   5   6 7     8  '
$ echo "$in"|awk -v n=2 '{ for(i=n+1;i<=NF;i++) printf("%s%s",$i,i==NF?RS:OFS);}'
3 4 5 6 7 8
However, all previous answers change all repeated FS  between fields to OFS. Let's build a couple of option that do not do that.
Option 4 (recommended)
A loop with sub to remove fields and delimiters at the front.
And using the value of FS instead of space (which could be changed).
Is more portable, and doesn't trigger a change of FS to OFS:
NOTE: The ^[FS]* is to accept an input with leading spaces.
$ in='    1    2  3     4   5   6 7     8  '
$ echo "$in" | awk '{ n=2; a="^["FS"]*[^"FS"]+["FS"]+";
  for(i=1;i<=n;i++) sub( a , "" , $0 ) } 1 '
3     4   5   6 7     8
Option 5
It is quite possible to build a solution that does not add extra (leading or trailing) whitespace, and preserve existing whitespace(s) using the function gensub from GNU awk, as this:
$ echo '    1    2  3     4   5   6 7     8  ' |
  awk -v n=2 'BEGIN{ a="^["FS"]*"; b="([^"FS"]+["FS"]+)"; c="{"n"}"; }
          { print(gensub(a""b""c,"",1)); }'
3     4   5   6 7     8 
It also may be used to swap a group of fields given a count n:
$ echo '    1    2  3     4   5   6 7     8  ' |
  awk -v n=2 'BEGIN{ a="^["FS"]*"; b="([^"FS"]+["FS"]+)"; c="{"n"}"; }
          {
            d=gensub(a""b""c,"",1);
            e=gensub("^(.*)"d,"\\1",1,$0);
            print("|"d"|","!"e"!");
          }'
|3     4   5   6 7     8  | !    1    2  !
Of course, in such case, the OFS is used to separate both parts of the line, and the trailing white space of the fields is still printed.
NOTE: [FS]* is used to allow leading spaces in the input line.