NRCan has a NTv2 web tool that can be used to transform coordinates between the NAD27, ATS77, NAD83(Original), and NAD83(CSRS) reference systems using the binary Grid Shift Files obtainable from the relevant provincial offices. You can use this tool to determine if it's worth going through the effort of tracking down and acquiring the GSB shift files from the provinces for your scenario. (None of the territories have grid shift files yet.)
From this example result for a point in nothern Alberta we can expect a shift of a of 25 to 30cm using NAD83(Original) to NAD83(CSRS) 2002:
+------------------+----------------+---------------+
| | Latitude | Longitude |
+------------------+----------------+---------------+
| Coordinate | 59.53642891 | -118.77687139 |
| Shift (metres) | 0.257 ± 0.172 | 0.288 ± 0.195 |
| Shift (seconds) | 0.008301" | 0.018341" |
+------------------+----------------+---------------+
At time of writing the available shifts in the tool are:
+-----------------------+-----------------+------------------+-----------+
| Coverage | From | To | Grid |
+-----------------------+-----------------+------------------+-----------+
| Canada | NAD27 | NAD83 (Original) | NTV2 |
| Alberta | NAD83(Original) | NAD83(CSRS) 2002 | ABCSRSV4 |
| BC (CRD) | NAD27 | NAD83(CSRS) 1997 | CRD27_00 |
| BC (CRD) | NAD83(Original) | NAD83(CSRS) 1997 | CRD93_00 |
| BC (Vancouver Island) | NAD83(Original) | NAD83(CSRS) 1997 | NVI93_05 |
| British Columbia | NAD27 | NAD83(CSRS) 2002 | BC_27_05 |
| British Columbia | NAD83(Original) | NAD83(CSRS) 2002 | BC_93_05 |
| New Brunswick | ATS77 | NAD83(CSRS) 1997 | NB7783v2 |
| New Brunswick | NAD27 | NAD83(CSRS) 1997 | NB2783v2 |
| Nova Scotia | ATS77 | NAD83 (Original) | GS7783 |
| Nova Scotia | ATS77 | NAD83(CSRS) 2010 | NS778302 |
| Ontario | NAD27 | NAD83(CSRS) 1997 | ON27CSv1 |
| Ontario | NAD27(MAY76) | NAD83(CSRS) 1997 | ON76CSv1 |
| Ontario | NAD83(Original) | NAD83(CSRS) 1997 | ON83CSv1 |
| Ontario (Toronto) | NAD27 | NAD83(CSRS) 1997 | TO27CSv1 |
| Prince Edward Island | ATS77 | NAD83(CSRS) 1997 | PE7783V2 |
| Quebec | NAD27 | NAD83(CSRS) 1997 | NA27SCRS |
| Quebec | NAD27(CGQ77) | NAD83(CSRS) 1997 | CQ77SCRS |
| Quebec | NAD83(Original) | NAD83(CSRS) 1997 | NA83SCRS |
| Saskatchewan | NAD27 | NAD83(CSRS) 1997 | SK27-98 |
| Saskatchewan | NAD83(Original) | NAD83(CSRS) 1997 | SK83-98 |
+-----------------------+-----------------+------------------+-----------+
NRCan doesn't seem to have a corresponding tool to demonstrate the generic nation-wide NAD83(Original) to CSRS conversion results.
This info gleaned from the links in @rgmrtn's answer to NAD83 to NAD83(CSRS) transformation for Northwest Territories, Canada and deserved to be fleshed out for a general not-locale specific answer. Do go vote that one up and read the things I've omitted.